Plumbing Noise – How to Eliminate Ticking Noise in Wall Pipe

Your options are limited being in an apartment, since you likely can't make very many permanent changes. I'll run through some ideas, though.

First, putting felt pads on the sides may help a little, but not for the reasons you think. They will do very little to dampen the vibrations -- you need mass or a dedicated damping agent for that. What the felt pads will help with is to reduce the mount of air flow through the door. Air flow is one of the absolute enemies of sound proofing. The more completely you can seal a door, the better it'll be. If you have the opportunity to attach an automatic door bottom (surface mount) and then some gaskets, then that'll go a long way towards quieting the noise.

Is your door hollow core? That is, does it sound hollow when you knock on it? If so, then you may look into replacing it with a solid core door. This can be done on a temporary basis as long as the hinges match up and you can simply pop out the old one and pop in the new one. A solid core door (especially a 1-3/4" thick one) will do a substantially better soundproofing job than a hollow core one. It does depend on you sealing it, though (see above) or else a lot of that benefit will disappear.

Beyond that... well, it really depends on how leeway you have towards making alternations to your apartment and how what your budget is. Attaching mass loaded vinyl to the walls and door would help, but would get pretty expensive very fast. Building an entirely separate interior wall would do a massive amount of help, but would absolutely cause noticeable marks when it was later removed. The flooring under the wall will certainly have to be re-done.

It's unlikely to damage your pipe, but the problem could be air in the system. You may have an old pipe underground that has a break in it.