Do press ups serve any purpose whilst building muscle?

You can make press ups harder by asking a training partner to load discs onto your back which provides more resistance. A push up is a gold standard in fitness training and I reckon it can help you build muscle. You can do

1 arm push ups
push ups with your legs raised
push ups with weights on your back

You should do them

Any bodyweight exercise will help you build muscle IF there is sufficient resistance which challenges the movement and is implemented into the right training programme.

You should be looking to fail between the 8-12 rep mark.

Using press ups as an example, Time under tension is a great way to make the most of this exercise. Make each rep 7 seconds long.

3 sec negative phase
1 sec static phase
3 sec positive phase.

Give that a try. Also try hand spacing. Work from the outside inwards.

Push ups will build limited muscle if used as the sole chest/triceps exercise because once you can do 50 or 60 push ups in a row or so (that's a fairly arbitrary number though), you'll stop building any more muscle or strength and just improve upon muscle endurance.

However, if you do a few sets of push ups after your more traditional chest workout with weights, then the push ups could lead to increased muscle damage and therefore increased supercompensation (when the repair process results in slightly bigger muscles than existed before the training session).

As Mark suggested using them as a superset is a great way to incorporate them into a program. Plyometric press ups or clap press ups can be used to increase power and therefore bench press or dumbbell press etc.