I have just bought some post mount calipers (Shimano) to fit on a cyclocross bike and separately some 160mm discs. The calipers do not quite sit flush on the post mounts (front or back) before the disc reached the back of the caliper. Should I expect to need washers to space the caliper out a bit further, and if so, what thickness?
It sounds like your post-mount frame (and fork) is setup by default for 140mm rotors. This would be expected for the frame and is not surprising for the fork either.
What you need is a post-to-post 140 to 160mm adapter. This might be more commonly a 160-to-180 adapter, but it's the same thing. It's moving the caliper ~10mm further away (or more like 12.5mm on top and 7.5mm at the bottom).
Shimano makes the SM-MA-F180 P/P2.
SRAM makes the "20S/20P" adapter also for this purpose. (This is the one I have used.) From SRAM's instructions publication, there is a handy table that illustrates that going from 140 to 160 uses same adapter as 160 to 180.
You might want to first check with a local shop to see if they just have the spacers you need in a bin. Otherwise you can definitely find it online (e.g. at jensonusa, but obviously it can be found elsewhere too).
Please note: while the question asks for post mount this answer pertains to the flat mount standard. Therefore, the following is not relevant to post mount brakes whatsoever.
Many Shimano brake calipers for road bikes are designed for 140 mm rotors. Two different types of mounting adaptors, fore and aft, may be used to mount larger rotors.
The rear adapter (eg SMMAR160DDA) is an aluminium bracket that adds 10 mm to the mount height through its own thickness and by moving the caliper a little forward.
The forward adaptor (eg SMMAR160DDA) places the brake mount further up on the fork.