I live in the Milwaukee metro area and we have a lot of houses on a well, my primary residence being one of them so here is my fairly well informed opinion on them.
Being on a well isn't a huge issue in it and of itself. The cost replace a well pump is usually around $1500 with labor and they have a life span of 10-20 years so it isn't a overly significant cost. The well pump isn't the only thing that can go wrong though. A well pump system also has a pressure tank (usually in the basement) which has pressure reliefs and cut off's that can fail. The tank itself can also fail and has a life span of 10-20 years also. The cost of a pressure tank replacement is usually $500-750. There also is a lateral supply pipe from the well to the house which can fail, this cost varies significantly based on the where the well is in relation to the house. The biggest concern with a well is that it can run dry or go bad in which case you have to either drill the well deeper or drill a new well. The cost of this can be vary from a few thousand to tens of thousands.
Like I said the well itself isn't a huge concern, and is something that can easily be accounted for in your capex calculations. The reason I won't purchase a rental property that is on a well is because in almost every case it also has a private septic system. This can be a major issue and come with significant costs. Private septic systems must be properly cared for. Of particular concern are famine products, condoms and plush toilet paper being flushed down the toilet. These can cause catastrophic failure of the leach field. Sending certain cleaning products and bleach down the drain can cause chemical imbalances in the septic tank that will also cause issues. If the leach field fails the system must be replaced entirely and because of new regulations this typically requires the installation of a mound system. This is assuming there is even enough land on the property to install a mound which has to be located in a different spot than the already failed leach field. A new mound usually costs around $15,000. If there isn't enough land then you must install a holding tank which must be pumped out usually every 2-3 months at a cost of $100-200. Speaking of pumping the tank even with a traditional or mound system most metropolitan municipalities require that it be pumped every 2 years at a cost of approximately $200-250. Passing this cost on to tenants would be difficult and generally unethical except in the case that they were in place during the entire 2 year period. It might be possible to prorate such an expense but would be much easier to pass this cost along in a slightly higher rent payment on the basis that there is no water/sewer bill.
For all these reasons I refuse to buy any rental property with a private well and septic. It won't command any higher rent but will significantly affect your capital expenditures. These added expense can be accounted for but that doesn't change the wild card of hoping your tenants don't ruin your leach field.