When someone travels to the US on H1B visa, is it mandatory to bring the I797 approval notice? Or is it okay to travel with just a photocopy of the I797 document?
Yes, it's mandatory.
USCIS lags behind going all digital, they still rely on checking paper documents. Whether it will get check or not is bit random, but I've checked with a few people who travel on H1 or L1 very frequently and they say that about 50% of the time they are asked for I-797. As far as I know, USCIS accepts only the originals.