Can an unbalanced-to-balanced adaptor damage an amplifier?

I didn't encounter any issues when I used the adapter XLR => RCA on Conrad Johnson GAT preamplifier and RCA => XLR on a Pass Labs amplifier. I don't think you have any issues but safe to confirm with T+A. 

I use a set of 4 XLR/RCA adapters for a piece with only XLR in/out, then I use my existing RCA/RCA cable to the Preamp.

I don’t hear any change/problems in the sound,

and I cannot imagine how they could damage anything.

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I bought cables XLR 6' long to RCA. Not much listening: medical testing period, in hospital 13 days, home at last: short time later, concentrated listening: what's this, an imbalance in my system? checked this and that, who would suspect the new cables. In the end, the guilty party was one of the new XLR cables. That's when I decided to use adapters with my existing nice/proven cables. Problem gone!

Another time I tried 25' long XLR to XLR to get audio out of an Oppo from Video system over to my music system. Added hum that didn't exist prior. Returned them. Used a crappy radio shack 25 foot rca/rca cable I had from long long ago: no hum.

I do like the positive hold of XLR connectors, but the majority of us don't need them in our Audio System. A slight increase of volume is possible using XLR, but not enough to use them.

Just check with T&A.

No problem on mine