When someone places a piece of paper or a tissue into a Siddur as a bookmark to save the place and keeps it there for a while, can that be thrown away? How long would that be to make a difference?

It would depend on the intention and the nature of the material used for the bookmark. If it is only temporary and it will be discarded promptly as usually a tissue is, it likely will not become an object of Mitzva. Not so if it is a more permanent and dedicated cardboard piece. However, one should not use a Sefer or Siddur as a place to store tissues which are going to be used mainly for one’s body cleaning, (or were used already).