Can I put the mat directly on a massage table or mattress or do I need to put a barrier between them?

The mat has special heat reflecting layers that direct heat and FIR rays upwards, which should be enough, unless you have a memory foam mattress or use the mat on the hardwood floor, in which case you can also put a towel, under-pad or a sheet under the mat.

If you use the mat with the cover (pillow case style) it also has a protective layer for under the mat.