Jesus says to cast your cares and your burdens on Him. His yoke is easy and His burden is lite, meaning that He will not become a burden on you or add to your burden, but just the opposite.
However just because we are to cast our cares on God, that does not mean letting go of responsibility. God calls us to be responsible people in all things. However it takes effort to learn to cast your cares, while not casting your responsibility. Learn to do this however, and you will be much better off.