Can renters install a doorbell camera?

Usually yes. Your lease is the first thing to check, then whatever rules the building has, and then your state's law on recording. Even if your landlord says it's fine, the building or the law still might not be.

An apartment front door on a shared corridor with clean, unmarked trim
The question is rarely whether a camera works. It is what your lease, the building and your state each allow.
Check your own state before you buy

Look up your own state's recording-consent rule before you buy. Most cameras let you switch audio off, which avoids the question entirely. If real money or a dispute is involved, ask a lawyer rather than a website.

If the lease or the building rules cover it, yes, and that is the reason to ask first. A camera that made no holes is far easier to argue about than one that did, which is why the mount matters more than the camera.
Often not, because nothing is altered and nothing is wired, but the lease decides this, and the battery does not. Ask about the mount. A camera held on adhesive is a different question from four screws into the casing.
This is the most commonly restricted case of all. A camera in a shared corridor records neighbors going about their day as well as your door, and managed buildings and HOAs frequently prohibit it outright even when the lease says nothing.
Adhesive comes away cleanly with patience and gentle heat; pulling it hard takes paint with it. Photograph the door before you fit anything, keep the original peephole and hardware, and take the mount with you as well as the camera.
Video of a public-facing area is rarely the issue, and audio is the part that catches people out, because consent laws for recording conversations differ by state. Most cameras let you switch audio off, which sidesteps the question. Check your own state, and ask a lawyer if real money is involved.