If you allow a hip or two to form early in the season on a remontant rose, will it continue to flower?
Or will that one hip make it stop blooming for the rest of the season?
No, allowing a few hips to set won't make it stop blooming. Some roses are more tolerant of this than others. Some will be slowed down by the presence of lots of hips, others don't seem to notice them much.
However, letting a few hips form may make you think about growing the seeds (which is actually a fun addiction).