if you do not trim up-keep clean as you go it is very important if you are sending it out- a long-armer may charge you more for the extra work of having to 'clean up' for you---i charge an extra $15 for this!
when left they can show through be quite unsightly- they can catch and cause tears, or seams to pull apart- don't go to all of the work to create a beautiful quilt top to leave it unsightly/messy ...to save having to do it all after it is done make it a habit to clip/clean up as you go= if you check & clip your threads on each block as you complete it then as you construct there is very little left when the quilt is finished.
i've had people bring me quilts they did not bother clipping- and telling me---'oh-don't worry about it-it will be fine' then when ugly threads show through in areas that are quilted= no way to get to them- they are upset (with themselves-not with me- who tried to warn them)
the back of your quilt should be as neat as the front! take pride in your work.