Firstly: 

If the bank is riba-based, then it is not permissible to deposit money in it except in cases of necessity, such as if he fears for his money and there is no Islamic bank or other place for storing his money safely in his city, in which case it is permissible to deposit money there, without taking interest. 

It says in a statement issued by the Fiqh Council of the Muslim World League, issued in Makkah in 1406 AH:  

It is haraam for any Muslim who is in a position to deal with an Islamic bank to deal with riba-based banks, either at home or abroad, because he has no excuse for dealing with them when there is an Islamic alternative available. He must turn from what is bad to what is good, and be content with that which is halaal so that he has no need for that which is haraam. End quote from Hukm wa Daa’i’ al-Bunook by Dr ‘Ali al-Saloos, p. 136. 

It says in Fataawa al-Lajnah al-Daa’imah (13/346): It is not permissible to deposit money etc in riba-based banks and other riba-based institutions, whether the account is interest-bearing or not, because that is cooperating in sin and transgression, and Allaah says (interpretation of the meaning): “but do not help one another in sin and transgression” [al-Maa’idah 5:2], except if there is the fear that it may be lost or stolen etc, and there is no way to protect it other than putting it in a riba-based bank for example, in which case he is allowed to do the lesser of two evils. End quote. 

Secondly: 

Some banks and riba-based institutions resort to a trick so as to promote themselves to people. They do not announce that they offer interest, and they offer prizes instead, for which they draw names or distribute lots at the end of the year or every six months. This method is used a great deal with regard to savings or investment certificates. This is a trick that does not make something haraam halaal. The bank does not supply the prizes from its own money, but from the interest, which it distributes in this manner, instead of distributing it to all its customers. This is a method that combines riba and gambling. 

Dr ‘Ali al-Saloos says in his book Mu’aamalaat al-Bunook al-Hadeethah fi Daw’ al-Islam (p. 38): 

Riba-based banks issue three types of certificate. The first, unlike the second, is aimed at attracting the largest number possible. With the second type of certificate they go further, because they take the total amount of riba and divide it into unequal amounts, and give them to the lowest number of depositors, then they distribute these amounts in the form of prizes which are awarded by drawing lots. So you may find a person who has deposited a small amount being given thousands of pounds, whereas one who has deposited thousands does not get anything.  So the former takes his share of the riba along with the shares of many others, and the latter loses his share to someone else. Every time the riba is shared out it is watched keenly, and one emerges happy with what he has gained whilst another grieves over what he has lost, and he looks forward to another opportunity. Is this not gambling? The riba-based banks have resorted to gambling with riba! The one who does not win his share of riba the first two times gambles with his share the third time. Isn’t the third type worse than the first two? End quote. 

The scholars of the Standing Committee were asked: Some commercial banks offer prizes such as cars or houses for people who open deposit accounts in the bank to keep their money safe. They draw lots among the customers of the bank, then one of the customers wins the prize. What is the ruling on this prize, whether it is in the form of cash or a gift? 

They replied: If the matter is as described, then these prizes are not permissible, because they are interest that is given in return for depositing money in the riba-based bank. Changing the names does not alter reality. End quote. 

Fataawa al-Lajnah al-Daa’imah (15/196) 

Thirdly: 

Once it is established that these prizes that are given by riba-based banks are the essence of riba and interest, then the one who has received any of them must get rid of it by donating it to charity. He must also withdraw his money from the riba-based bank, except in cases of necessity, as stated above. 

It says in Fataawa al-Lajnah al-Daa’imah (13/354): Riba and interest are haraam forms of wealth. Allaah says (interpretation of the meaning): “Allaah has permitted trading and forbidden Ribaa” [al-Baqarah 2:275]. The one who has acquired any such wealth must get rid of it by spending it in ways that will benefit the Muslims, such as building roads and schools, or giving it to the poor… End quote. 

Fourthly: 

Money that is deposited in the bank, which they call “deposits” or “current accounts” etc is in fact a loan that is given by the owner of the money to the bank. As that is the case, it is not permissible for the lender to take anything from the borrower in return for that loan. The scholars are unanimously agreed that every loan that brings a benefit is haraam. 

Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (6/436): 

Every loan in which it is stipulated that more be given back is haraam, and there is no difference of opinion on this matter. Ibn al-Mundhir said: They are unanimously agreed that if the lender stipulates that the borrower must give more back or give him a gift, and he lends it to him on that basis, taking the extra payment is riba. It was narrated from Ubayy ibn Ka’b, Ibn ‘Abbaas and Ibn Mas’ood that they forbade loans that bring benefits. End quote. 

And Allaah knows best.