Can I replace MC33204P with one of these other op amps?
The LM339 absolutely not. That's a comparator, not an opamp, and it has an open collector output.  

The LM348 is a quad opamp like the MC33204, but has the specs of the LM741, and this is an age-old opamp with specs inferior to the MC33204's. You could probably replace the MC33204 with the LM348, but don't expect the same performance. An opamp has many parameters, and it's not easy, comparing two opamps, to pinpoint which one in a specific design will be better or worse overall. So, while the LM348 may work (just try it), I would stick with the original if possible.  

edit
On second thought the LM348 may be usable. I only had a brief glance at the schematic and only saw that it worked with audio signals. Missed the LED color organ. For this you don't need a Hi-Fi opamp, like you would if the audio would ultimately go to a pair of speakers. The LM348 will be OK.