The bug
Carpet placed next to a door prevents the villagers from entering or exiting, this causes them also to sometimes get stuck into a loop opening and closing the door. Villagers moved freely in 1.8.9.
Not sure which snapshot version this started although I suspect it was when villager height was increased in 15w39a or possibly if carpet properties changed.
More specifically, all mobs of height 1.95 or greater cannot move onto carpet if they are currently inside a 2-block high area. This affects:
Drowned
Evoker
Husk
Illusioner
Piglin
Pillager
Villager
Vindicator
Wandering Trader
Witch
Zombie
Zombie Villager
Zombified Piglin
Skeleton (1.99)
Stray (1.99)
Prior to 1.9, villagers were 1.8 tall, and could step onto carpet freely, but other mobs still were affected by this behavior. Note that mobs still try to pathfind onto the carpet, even though they fail to move through it.
Related issues
is duplicated by
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The height of a villager plus the height of a carpet (or even a lily pad) is now more than 2 blocks. That is why they do not fit. I doubt this is any thing other than WAI. However, villagers still try to path through a space through which they do not fit. That is probably a more appropriate framing for this issue.

I can confirm it happens on 1.9 release. I've got a whole village of villagers constantly opening and closing doors they then can't get through. Its... loud.

Confirmation for 1.9.2:MC-100323

MC-97042 was updated to include carpets

Reopening, this is separate from the AI bug.

Like @Nathan Ryan said, this has to do with the heights of the villager and the carpet stacking, so that it prevents movement.

I attached screenshots, so you can see how a carpet affects an entities' Y Level.

To fix this, I think the height of the carpet should be changed.

The AI bug is that the Villager doesn't ignore this invalid house, this issue describes what a carpet does when placed next to a door.

I found a solution to this that mojang needs to do. It is to make the collision of the carpet the same as the snow layer block, when the snow layer block is floating, you don't stand right on top of it, you sink through it a bit right to the bottom part of the block.

Please do not mark unreleased versions as affected.
You don't have access to them yet.

Affects 18w31a.

Affects 1.13.1-pre2.

Hey. So if This is an old bug that hasn't been patched since 1.9. Why are they making the New Village buildings as if it were already patched?
[media][media]
Confirmed for 19w13b.
This relates to MC-50367

I'd like to applaud @Jessedward for her dedication in keeping the affected versions for this bug updated over three years.
Although they didn't fix this, they did fix carpets blocking the exit of the village.
Edit: I have recently noticed that there may still be issues with this in some villager houses

Placed a carpet next to the door and now villagers cannot leave. Bug confirmed in 1.14.1 pre2.
Confirmed for 1.14.1

Still present in 1.14.4 full release.
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Happens in 19w42a

It is caused by the change in 15w39a (height changed from 1.8 to 1.95 blocks tall). It should be reverted or carpets should be the same as 1 snow layer because this is very very annoying.

Can I request ownership, so I can keep this bug being updated? Confirmed for 1.15.2 and 20w11a

Confirmed in 20w12a

Still present in 20w13a.

Affects 20w13b

Affects 20w15a

carpets have a hit box so if the hit box of the mob is larger than the space - the carpet hitbox then its the same as walking onto a slab, the hit box is different however
also if you shift then you should be able to get through

You can get through it even if you are not sneaking.

put a trapdoor above you in the top half of the second block up so its in the same space as your head, then place a carpet infront of you and you should not be able to get through with out shifting

This is not resolved. Still does it in 1.17.1 Java version on MAC.

It is. It's marked as Working As Intended.

hi, just wanted to add that this bug is still happening in 1.21.1? and it feels like it shouldn't work like that, especially since several villages come with carpets in them now. i just found a villager in an ice biome who couldn't get to their jobsite because there was carpet in the way. they can walk on them if there's 3 blocks, but not 2? i would like it if this was fixed. it feels a little bit silly
@unknown This issue is resolved as Working as Intended, which means it's not a bug and won't be fixed.