I dont think there is a clear yes or no answer to this question. Like most things, it depends on other factors.
If the coach is using a form of RBBC or uses specialists in receiving roles or short yardage roles then there is actually more than one "starter" for those roles. The team has back ups and contingency plans for if the starter goes down that doesn't change the plan much. How many touches the RB gets depends somewhat on game script. If the team is winning then "the starter" gets the majority of the opportunity with perhaps that players back up coming in later in the game to mop up. If the team is losing then the receiving RB may end up playing more snaps than the starter does and thus more opportunity for the "back up" than the starter in such situations.
To give an example the Washington Redsnkins have Adrian Peterson, Derrius Guice in what I would call the starter roles. They also have Chris Thompson and Bryce Love for the receiving RB role. If Peterson gets injured I don't think that would have any effect on Thompsons opportunity but it would have a huge impact on Guice. Conversely if Thompson was injured I don't think it would mean more passing targets for Peterson, but it could mean more for Guice who might slip into that role then. If not then it would be Love as the direct back up to the COP role that Thompson has.
Why?
Mostly because Peterson is not a very good receiver. Guice might not be either. We don't know that yet. I think he could be better as a receiver than Peterson is though. I could see him taking over that role in the offense in such a scenario where Thompson and Love are unavailable (I don't think Love will be active this year??) which is only a Thompson injury away from happening. Perine is in the mix for the "starter" role and may be the odd man out as no room for him. If he does make the team and Peterson/Guice are out then it wouldn't be Thompson stepping up for a bigger role but Perine coming in.
Every team is different. The coaching philosophy and their personnel. Scenarios like this exist for every single team though. Most are a bit muddier than the Redskins situation I think, that is why I used them as an example.
If the back ups are really bad then teams will use more desperate measures such as moving one of their WR to RB like the Packers did with Montgomery awhile back. Some teams do have players bad enough that they never want them to see the field but there is always a free agent out there who could come in and contribute instead. See CJ Anderson last year. So I think it is a pretty extreme case for the back up to be so bad the team just decides to give their starter all the carries. That is poor risk management and planning. They should always have back up plans.
When the starter is very good its just hard to take that player off the field. Especially if they are versatile and not game script dependent. I don't think they are playing more just because the back ups are bad. Maybe in some exteme short term situations that might happen, but I don't think that is ever part of the coaching staffs plan.