Hmm...
I have CS4 - well, at least I did.
I upgraded to Sierra, on my main drive which had a working CS4 that I did not disturb for the Sierra upgrade. CS4 worked fine for a few days.
I think there was some issue that came up, maybe importing a file that produced an error...
Anyway, I decided to reinstall CS4. Did not work, could not reinstall CS4 on Sierra.
So, an already-installed CS4 may have issues, but might work for you. Wouldn't reinstall (the installer crashed with some kind of error).
And, that's CS4 - a year or two newer than CS3.
Good way forward is to test that for yourself. Take a drive that has your CS3 installed and working.
(This is a drive that you can test, that's backed up, and that you can restore to the older system if this doesn't work out for you)
Upgrade that drive to Sierra. Try Photoshop.
I can't tell you what may work, or what will not function at all. Could be that you will be fine with 10 year-old software, or you will discover that you can't fix some issue - unknown on software that is at least 4 version behind even old versions that might still be supported by Adobe.
Bottom line, I had problems with CS4. I don't expect that CS3 will be better. It will likely be worse.