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If i have a 10.1.0.0/20 network allocated to me, and i have three routers, do i have to use addresses within this address space for my loopback interfaces on the three routers?

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It depends on what you're doing with those loopback interfaces.

If you want them to be routable outside those three routers, then yes, you'll need to use your allocated space.

If it's just those three routers you want to talk using those loopback interfaces, then you don't need to use your allocated space, you can use any IP address, but stick with "private" addresses just for sanity's sake.

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