Is it permissible to put money that has been found, and its owner could not be identified, into the Zakat fund after ten days, and is this considered a sin?

If the found item is valuable, it must be announced for a full year in places where its owner might be found. If it is more than negligible but not excessively valuable, it should be announced for a few days. Placing a found item in the mosque's Zakat (definition) box is a dereliction of duty on the part of the finder, who is obligated to announce it. If the owner comes and approves of the finder's action, then all is well. Otherwise, the finder must return it to the owner, and what was placed in the box will be considered a charity from the finder.