I have a Viewsonic PJ558D projector. I replaced the light bulb recently and have not been able to use the projector since. When I turn on the projector, the fan turns on, the bulb lights up, then everything shuts down after a few seconds.

You need to reset the lamp timer. This is likely your problem since you changed the lamp.
Turn the projector on. open the menu. Go to advanced. Scroll down to the first option which should be auto source. Make sure this option is checked. While on this option press right once, left twice, right 3 times, then left 4 times. Another menu will appear. Scroll to lamp time and push enter. This will take it back to zero.

It sounds like the cooling air path is being blocked somewhere, the cooling fan is not operating properly or the air filter is already clogged with dust. Remove the air filter and visually inspect it. Bang it on a flat surface with the outside surface facing down. If only a small amount of loose dust falls off, then you have additional cooling problems as I mentioned above and you should return it for warranty repairs or exchange. Operate the projector without the filter and see if the problem returns. If not, and the filter looks ok physically I would insist on an exchange or warranty repair.