The sound of rain causes the game to stutter when moving at high speeds (such as in a minecart or a speed potion). Moving far enough away so you don't hear the rain (example: going underground or entering the nether) causes a big frame drop for a few seconds and then the game returns to normal.
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Are you sure its not all the rain particles causing the lag?
Now that you mention it, no I'm not. Disabling the particles probably would help, but only because (unless something has changed) the sound is tied to particles.
The indoor rain sound causes more stuttering than the outdoor sound, but the difference isn't that noticeable. I've also been high up on a monorail and still had the issue.
However to try to narrow down if it's the particles or the sound, tonight I will try to create a control by creating a situation with a comparable number of particles and seeing if running around causes stuttering.
Confirmed it is the sound.
1. Gave around 200 villages potion effects and rode around in a minecart. No stuttering.
2. Rode a minecart in the rain with sound @ 0%. No stuttering.
3. Rose the same minecart in the same direction in the rain with sound. Game stutters.
DarkWolff: Can you check the performance graph (Shift+F3) and tell us if you see a big difference on the "Sound" section when it's raining?
When it's raining and I'm moving at a normal speed it's basically the same as when it's not raining. When moving really fast (full speed minecarts or running + jumping), at the same time the stuttering starts both the sound and ticks increase in percentage a lot in between stutters. The tick seems to be going up faster however.
This is a notorious issue with rain, but it's also caused ocassionally by other sounds. I worked around it by deleting the OpenAL files in bin/natives and installing them in my OS. Never had sound lag problems again.
I followed the steps from here: http://www.minecraftforum.net/topic/892045-sound-lagstutter-fix/

I've investigated this issue in the past. It appears to be caused when the indoor-rain sound effect stops playing. So going in to a house starts the indoor sound effect and then going down into a mine stops the sound from being heard, that's when the major stutter happens.
Since the game uses the same actual sound effect for both indoor and outdoor rain, looks like the problem might be in the pitch-adjustment code. (Making a quick mod to make all rain sound the same fixed the stutter, but at the cost having the same noise inside and out).
Hope that's at least slightly helpful.
This seems fairly likely as the issue isn't as bad if I'm in a minecart above ground. The issue also intensifies when I move far enough away for the indoor rain sound to stop.
Finally, I recently had a similar issue with a few enderman growl sounds playing at once effectively freezing my game up long enough for them to kill me. It was an isolated incident, but still notable in that it matches your hypothesis.
It's not just rain. When ever the play move away from any continuous sound source (such as lava, flowing water, rain, etc), minecraft will have some sort of heart attack. I hate it.
From the looks of it this is the same problem as MC-560. I've posted a fix in the forms of a mod in that bug report, as well as a description for Mojang about how to fix it. Maybe someone can verify that it's caused by the same bug by installing it?
Thanks Sopa, your fix worked, minecraft is running great again!! Reposting link here:
http://www.minecraftforum.net/topic/892045-sound-lagstutter-fix/
Is this still a concern on Minecraft 1.4.4 Pre-Release (without any modded client)? If so, please update the version affected.
Updated to add 1.4.4 to the list.
Doesn't appears to be caused by the sound.
Can you state your reasoning for this? Turning off the sound causes the issue to disappear, so it's likely a part of the sound engine causing it from my point of view.
Sorry, I didn't read any of the previous comments... I read them now. My screenshot shows that most of the processing capabilities were occupied by the tick, not by the sound. But if turning off the sound stops the problem, so the sound is jamming the tick...?
I've noticed something similar, but the sound percentage goes up as well; the tick just goes up faster.
I've noticed the issue in two places around my main residence. The first is as I approach my front door - next to it is a pen with many cows and sheep. Just inside is a large room with many torches. Sound stutters with a severe drop in FPS which eventually resolves after a few seconds.
The second place is very nearby my residence in a woods I planted. In this case, there aren't as many animals (other than perhaps a few chickens).
The issue seems to have started recently - not associated with any release but coinciding more with perhaps my chicken farm or the enlargement of my main room in my main residence (from 2 blocks to 5 blocks high and larger depth/width).
The issue is tightly tied to the two locations and occurs every time I approach them. I'd love to try the OpenAL fix, however, the link provided in the mincraft forum (http://connect.creativelabs.com/openal/Downloads/Forms/AllItems.aspx) now apparently leads to a broken page. If anyone has a suggestion of where to obtain the 64-bit OpenAL drivers, I'd appreciate it. Google seems to always lead me back to this broken page. Oh, and attempting to report the broken link using the "Contact" page on Creative Labs' web site results in... a 404 error. Jeez.
In the meantime, I may try slaying my chickens and cattle to see if the issue is indeed resolved with a reduction in noise.
EDIT
I slaughtered all my chickens but left my cattle. The issue was resolved completely. I then introduced a few chickens to my coop again and, wouldn't you know it, the issue came back. It also seems proportional to how many chickens I have, giving weight to the idea that at some point more sounds == more stuttering.
Suggestion - add an option to "clip" the number of concurrent sounds so that those of us with under-performing audio devices might be able to mitigate this issue.

Confirmed for 1.4.6
Confirmed for 1.4.7 AND that the fixes suggested in the comments in MC-560 solve the problem.
Lag/stutter is more noticeable on weaker systems.
Mojang please review this as a priority as it hinders game-play significantly on some systems.
It's more the sound card than the system in general. I got a new sound card last week and the problem has diminished, though still noticeable. My game doesn't grind to a halt anymore, but it does stutter somewhat.
This same issue is caused by moving around large quantities of lava, like in the Nether or some custom maps, etc. It seriously hampers the usefulness of lava.

Can we get a confirmation of whether this happens in 13w10a and an update to the affected versions?
@Chad Anderson, it still happens since I'm getting lag with the lava on a 1.5 server map. Haven't checked about 1.5.1
I've posted a pic of another way that this lag can happen and it's been happning since at least 1.2 or 1.3 maybe.
How to reproduce:
1. Make a 2x2x2 hole.
2. Fill with cows (pigs and sheep should also reproduce it) to at least 100+
3. Walk away from the cow pit and you'll have sudden lag.
It happens when the sound is set to on an doesn't happen when it's off. Also doesn't happen when approaching it.
Happens with other continuous sound sources such as flowing water, lava, and splashing. The sound system really needs fixed.
I too have been having same issue. I bought a new sound card, updated all drivers and Java. Nothing seams to fix this issue. As GerbilCrab475 said it happens with all continuous sound sources, Mine cart, lava, flowing water, etc. Please fix.
I consistently get this error, too. It also appears to interrupt with being able to catch fish when fishing underwater (to escape annoying rain noise & increase #fish coming.) I will often hear the fish, but as I go to get it, it will not come. I am convinced this is due to the HUGE drop in FPS that comes with rain. I think this error is a kind of 'stuttering' – of fish catching rather than sound (though I get that, too).
I hope this additional info helps you in debugging this issue.
The sound of the lava causes more lag

Cannot reproduce in the latest version of minecraft
Can confirm for 1.13.1

Can you make a video? This was resolved as "cannot reproduce" over 5 years ago.