Does a security deposit placed in an interest-bearing savings account and registered in the tenant's name constitute usury (riba), which is forbidden?

If the deposit is placed in a usurious bank, it is a form of usury, and you must avoid any contract that necessitates it, unless you cannot find another apartment with a contract free of usury. In that case, there is no harm, and the sin is upon the one who compelled you. If the deposit yields interest, then spend it on charitable causes and for the benefit of Muslims.