Aaron Carroll had a very useful NYT Upshot piece highlighting research showing that even modest co-payments discourage people from getting necessary medical care. The article is about co-payments for prescription drugs where it highlights research showing that people will often skip taking prescribed drugs to avoid co-payments. There are a couple of points worth making about co-payments in this context and more generally.
First, if a drug has been prescribed for a patient, then it is the judgment of a medical professional that they need this drug for their health. The argument for co-pays, that we want people to think twice before getting the treatment, really should not apply here since a medical professional has determined that they do need the treatment. It doesn't make sense, in general, to encourage people to substitute their own judgement for that of a medical professional. (That doesn't mean that medical professionals will always be right, but it would be best if patients made the determination to ignore their judgment based on their own research, not the desire to save a co-pay.)
The other point is that drugs are almost invariably cheap. By this, I mean that they are cheap to manufacture and distribute. The research can be expensive, but this is a sunk cost at the point where the drug is being prescribed for the patient. If all drugs sold as generics, with no patent or related protections, they would rarely cost more than $10 or $20 per prescription. For this reason, there are not much savings to society if we get people to take fewer drugs, we are just risking people's health with co-pays.
We do need to pay for the research. I suggest doing this upfront, with the government contracting out to private firms. All results and patents are in the public domain. (See Rigged, chapter 5 [it's free].)
The issue with drugs is qualitatively different than with doctors' visits. First, a visit to the doctor does require the use of a doctor's time, as well as the time of their support staff and possibly other health care professionals. This means that there actually are savings from discouraging unnecessary visits.
The second point is that a decision to visit a doctor depends on the patient's judgment, not that of a medical professional. (This is not the case with repeat treatments, just an initial visit.) People will always weigh several factors in deciding whether to visit a doctor, whether or not there is a co-pay.
For example, if they have a busy schedule or long-planned travel, they may choose not to see a doctor for a particular issue where they might otherwise see a doctor. In this context, it may be very reasonable to have a modest co-payment (e.g. $20 per visit -- which would be waved for low-income people) to make people think twice before seeing a doctor.
This sort of co-payment can be seen as analogous to charging people five cents for using a plastic bag when they shop, as many cities now do. If someone really wants the bag, the five-cent fee will not prevent them from getting it, but it does get people to think twice, and therefore has led to a large decline in usage.
It is reasonable to think that a modest co-pay could have a similar effect on doctors' visits. It should not prevent people with serious health issues from seeing a doctor, but it may discourage some visits for relatively trivial matters, like a cold.
Note, this is not the "skin in the game" story pushed by many economists, which wants patients to be comparative shoppers. There is considerable evidence that patients generally are not good at weighing the relative price of different treatments, and when they do, they often make the wrong choice for their health.
This is simply arguing that it would be good if patients think twice before rushing to see a doctor. We can't guarantee that this will never mean that a person who needed to see a doctor chose not to, but in the alternative, we can't guarantee that a person in desperate need to see a doctor won't have to wait because the person in front of them in line has a cold.