My car is kind of a beater and I don't want to pour money into it. Apart from a bumpy ride, is anything really bad going to happen if I don't replace these parts? For example, will this put a significant amount of extra stress on the motor mounts?
There are two big things which are a detriment to riding on worn out struts and bushings:
- Safety: Worn struts cause longer stopping times/distances as dead struts allow the vehicle weight to shift (sometimes unexpectedly) during braking. This weight shift is a lot more than when you are using good struts, so braking distances suffer. You also will not be able to corner as well as good struts will allow you, either. You will notice both of these effects if trying to do either over bumpy roads ... the struts will just not be able to allow you to control your vehicle in a safe manner.
- Wear on other parts: The most notable things which worn bushings and struts cause is accelerated wear on the tires. Because the bushings hold the alignment in check, with the tires at the proper angles, when they wear out, it causes extra wear on the tires. Worn struts also allows the vehicle to bounce more, which causes wear on your springs. Springs are meant to support your vehicle's weight, not to bounce uncontrollably up and down.
I understand not wanting to put unneeded money into your vehicle, but sometimes if you want to continue to use your vehicle, the smart thing is to replace the worn parts. This doesn't mean you have to replace them with upper tier replacement parts (such as buying them from the dealership), but you should at least get them replaced with whatever you can afford.