Is it OK to work on projects from research proposal before position starts? There is considerable gap in time between submitting applications for academic positions and the time the postitions start. I'm wondering about the status of the problems listed in the research proposal during this gap. Are they off-limits and should be avoided, or is it OK to work on/think about them in the meantime? Suppose that I show up at the new institution already with a preprint on arXiv solving one of the problems from my research proposal - would it be considered inappropriate?

[My field is pure mathematics, meaning that the research tends to be relatively unpredictable and does not require external resuorces.]
I don't think you should ever delay work, especially in mathematics. It is like trying to turn off your brain. Work away. I'd even say publish all you want. You aren't being hired for specific pieces of work unless you are a consultant or such. But academic positions are nothing like that. 

The "worst" case is that you show up with everything completed and have to - horrors - work on something else that extends it or is otherwise "interesting." 

Kidding aside, nothing is off limits in advancing mathematics. You will only be praised for working in the interim. Your research proposal just assured the hiring group that you have ideas and will be productive.