Things like the PzKfWg 38(t) would of course not cause too much strain, since the Germans captured quite a few as well as the factories too. But what about things like using captured artillery and other tanks?
I know the Whermacht used a lot of captured French artillery, as well as Soviet artillery too. The Japanese surely captured many weapons as well.
Did they, or any other Axis powers, strain their logistics by using enemy weapons? They surely would have a hard time at least starting the manufacturing of ammunition and even getting spare parts.
In short, no, not on strategic level. Captured Czech equipment was instrumental in German victory in Poland and Germans were experts in reusing captured equipment. Examples are Panzerjager I, Marder I etc. However, as war progressed larger and larger portion consisted of German technology. Spare parts and ammunition could definitely have been a big problem, but on a local level. German divisions on the move consumed around 300 tons of supplies per day and with operations routinely involving 30 to 100 divisions supplying them was creating much greater logistical difficulties. Also, in 1943/1944 germans were pumping tens of thousands of both tanks and planes, usually superior to anything captured (with the exception of t-34) which was far outweighing quantities of captured equipment.