I'm quilting (longarm) an appliqued quilt at the moment that has this kind of stabilizer. I tore off the bigger pieces from the back, but left smaller ones ones in. So, some of the quilt has stabilizer, and some of it does not.
I haven't noticed problems stitching through any parts of the quilt, or moving from stabilizer areas to areas without it. I would think that, as long as your quilt top is nice and flat, your longarmer shouldn't have any problems with it. If you lay it out and notice any areas that are bulging or otherwise uneven, press them flat using starch and perhaps some steam.
Be sure to post the finished quilt!