I find myself in a bit of a predicament. My PhD funding ceased this month, and so I started applying for jobs from June- literally, doing anything! One job was in a University, in a support role, excellent salary and permanent. I applied for this in July, forgot about it, and then was offered some hourly-paid teaching from my PhD university, which I accepted. This is only a few hours a week and so I have had to take some bar work to supplement my income, whilst I finish up the PhD (I'm fairly near the end). I have also been somewhat messed around regarding the teaching; I cannot get an answer as to when I will be paid, and the hourly rate has been lowered, although nobody can tell me what to.
I have now been offered an interview for the other university, in the support role, and attending would mean missing my hourly paid teaching, which will be difficult to re-arrange. Also, there is no definite start date- it can be decided if I am successful- but I am unsure what to do if I get the job, and they wanted me to start before the teaching finishes.
The last thing I want to do it upset my current university, although they've not treated me too well, but well paid and full time university positions are hard to come by. Any advice is much well-received!

I've thought about being in this situation when I finish my PhD since I made need to get a job but also want to maintain contact with my university. I thought I would take the fulltime job, but book holiday, work different shifts or call in sick at the job if I had to, in order to accommodate my uni work.

Since the work at uni will only be for 3 - 6 a week and not every week, I think that this should be achievable to do both. Alternatively if the work were to fall regularly, I'd just ask to change my working pattern or reduce my hours at the job.

If I were you, I'd take the job, especially as you don't know the start date anyway. You can work out what you are doing to do about working it later.

There is no predicament here- go to the interview!

First, is that the only possible time for the interview? Maybe explain when your teaching comittments are, and ask would any other time be possible (but say you can make arrangements for that time if necessary).
Second, if you need to miss the teaching, then miss it- these things happen, people get sick etc, make up a reason or call in sick if you don't want to tell them about the interview.

As treeoflife says, if you get the job, you can figure out how to deal with the teaching work then- people leave jobs, its not that big of a deal, or the new job might allow you to delay or alter working hours to finish the teaching.
Good luck :)