Should I cut this tree and remove it?

cutting a tree down will not remove the roots. Sometimes you can do what's called a root trace and dig one or two roots out without removing the tree. Large roots anchor the tree. Small roots feed the tree. Obviously if you cut a couple large roots you will also lose feeder roots. It can be done but depending on the situation you might harm the tree ...I was offering as an alternative to totally losing the entire tree. You could build up the soil area also...but be careful not to push soil up against the trunk as this can cause other problems.

Susan is correct, the landlord owns the property and I would assume your lease does not provide for a tenant to remove trees from a yard. Please keep in mind that if the landlord does try and remedy the root situation by covering the roots with soil the trees will eventually die. Also by chopping our the roots the trees will also die over time. So if the trees are a problem, removal would seem like the best options. If the landlord does remove the trees for you and if they desire to plant another tree here is a link for some fast growing trees: http://www.naturehills.com/catalog/Trees/Fast_Grow...

You may not remove trees from your landlord's property without permission to do so.

If you think the tree's roots are a hazard you should ask the landlord to come up with solutions to the problem. If the roots truly are a hazard the landlord should pay for any necessary measures.