There are no "D5100 lenses" In Nikon speak there are DX lenses and FX lenses. Tamron FX = Di (not Di-II) and Sigma calls FX - "DG". In any case, if it is an Nikon F mount it will mount and work on both the D5100 and D750.
As has been said above, just because you can does not mean you should.
When I added my D610 I continued to use some of my DX lenses I had for my D5100. I don't anymore as I now have FX lenses more suitable. But until you add to your collection of full frame lenses you certainly can use the lenses you have for your D5100 on your new D750. It won't hurt and you will get good images.
You will want to add FX lenses as you are able. I started by buying some used (err vintage) lenses to get started. I've since started to add more expensive newer models. Nikon lists the compatible lenses here:
https://support.nikonusa.com/app/an...t-lenses-can-i-use-on-the-nikon-d750-camera-?
There are really old 'non-ai" lenses that you need to avoid, some can damage your camera. These are true antiques.
If you do not already have a 50mm full frame, start there, they are so versatile. The AFD 1.8 can be had for about $130 (just checked B&H prices). That is the so called "normal" lens with the 'niffty fifty' nickname. If you can afford a 1.4 or the Siggy Art version even better but the AFD 50mm 1.8 is probably the best bang for the least buck of any lens.
That link to Photographylife mentioned above gives a great explanation of the differences.
Enjoy that new D750 and don't be afraid to use any lens on it.