Is it permissible to deal with savings accounts in usurious banks, whereby the bank saves a monthly amount from one's salary with an annual increase, if this increase is disposed of by giving it to the poor?

It is not permissible to save in usurious banks, and the interest they give is considered unlawful usury (riba). The intention to get rid of it by giving it to the poor does not make the forbidden transaction permissible. One must deal with Islamic banks if possible, and a Muslim has no excuse to deal with usurious banks when an Islamic alternative exists.