I know there is something like adhesive tape but it will not leave marks and is designed not to leave marks on paper. Ordinary tape is highly adhesive and could easily damage prints on paper if not removed carefully. What is the name of such thing?

A Post-it note?

https://www.staples.co.uk/office-supplies/writing-supplies/correction/post-it-cover-up-and-labelling-tape-25-4mm-658h

Painter's tape. Over here, it's blue and has the feel of masking tape, but you can use it to mask what you don't want painted and later peel it right off.

There’s different levels of adhesion depending on which painters tape one gets, and some have different textures that work better on various surfaces (smooth versus rough, for example). The paint section of your local hardware store has lots of educational opportunities. Even different brands of post-it’s have different levels of tack. 

For the paranoid: artist's or drafting tape.

There’s also acid free tape one can use for scrap booking and which is safe for photos and portraits and for art. Scotch and 3M are here too common brands