Should I got to a different dentist as I was told my front teeth gap won't completely close? (Photo)

After have reviewed your photo, I don’t see why Invisalign wouldn’t close your gap. Your teeth are shaped correctly and Invisalign shouldn’t have any issues to close your gap. Go see another dentist, redo another digital scan, and ask to see the result simulation.

In most cases, gaps can be closed predictably by moving the teeth either inwards or sideways towards each other....or a combination of both.  It all depends on your situation. There are a few of reasons why a gap may not close.  For example:  1) the overjet is too tight or 2) a tooth is ankylosed or frozen to the bone.  There are ways to get around these situations, but it may require more advanced treatment than aligners done by a non-specialist can achieve.  Remember it's not the paint brush that makes great art...it's the artist.  The same applies to orthodontic care.  I highly recommend you visit an orthodontic specialist who is trained and experienced to recognize and asses these types of situations and can deliver you the awesome smile you deserve!

Based on your photo I would ask to SEE the clinchek your dentist got back.  If there is an issue with closing the space, there will be notes submitted with the case as to how or why the space won’t close. It may be that they promised it to be done within a certain number of aligners. A longer case may fix the problem. 
If they won’t show you, get another opinion