Overview.

This is a very interesting question. The ideas behind this have been around for a while and the methods are covered in great detail elsewhere.

The overall answer to the hypothetical question is rather surprising. There may be actually potential benefits to perpetual liquid ventilation for people with certain diseases. Trials so far have not found side effects in fluorocarbon liquid breathing.

So lets talk about the history of liquid breathing and case studies, then we will move onto the hypothetical situation of liquid breathing indefinitely.

A great review (Shaffer et al, 1992) covers the field in great depth.

@MCM pointed out in the comments that my original answer did no emphasise that animal diaphragms are not powerful enough to move the liquid in and out of the lungs. I should mention now that the below cases all use ventilators to pump the fluid into and out of the lungs. The difficulty of breathing this without a ventilator is emphasised by David Blaine at 4:54 in this TED talk who said it was like trying to breathe with an elephant standing on your chest.

Case studies.

As early as the 1920s, poisonous gas research that lead to the infamous mustard gas found that lungs could tolerate large quantities of saline briefly without damage.

Animals.

Kylstra, the first pioneer introducing the idea of land animals breathing liquids (Kylstra et al., 1962), found mice could withstand 4 hours of 160 atmospheres of pressure.

Clark found that mice and rats could breathe oxygen bubbled silicone oils and survive for a few hours after emersion.

In 1965, he discovered that the fluorocarbon known as FC75 not only supported respiration during total immersion but the mice survived indefinitely in apparent good health.

The most recent study I can find is the partial use of fluorocarbon liquid ventilation in adult respiratory distress syndrome (ARDS) of rabbits. The same improvements in breathing were not observed in saline liquid breathing or conventional ventilators.

Humans.

In humans PFCs were first used successfully in 1989 for the ventilation of young infants near death.

Marked improvement in lung compliance, without a compromise of cardiovascular status was noted in all three patients.

All three infants died of their severe preexisting respiratory complications, but the trial showed liquid breathing can support gas exchange in humans. However developing lungs of infants cannot rely on mechanical ventilators for prolonged periods as nearly a quarter of premature infants of those that do later developed chronic breathing problems.

Biomedical application.

There are a fair few researches and doctors who believe fluorocarbon liquid ventilation has some advantages over gas ventilation. It reduces stress at the area of interfacial surface tension in the distressed, surfactant-deficient lung. This would be ideal and appropriate for ARDS in human adults. A more sophisticated use would be using fluorocarbons as a drug delivery system straight to damaged lung tissue.

(All the above can be found in the 1992 review. There is also data on the different properties of suitable FCs)

Real world potential.

Is perpetual liquid breathing possible? ...And healthy?

So this is where we stand with the case studies; mice can breath liquids indefinitely and stay in good health, rabbits with ARDS survive where breathable gas would not help, and there is a contingency in the biomedical community that it can be better than gas ventilation for medical treatment. To me it seems that there is nothing to suggest a fully grown adult would suffer from breathing liquids. The inertness of the fluorocarbons implies any toxicity would only reveal itself in a timescale of years, and everything else about the compounds pose no danger and may make gas transfer across the lungs easier.

This relatively modest success is perhaps no surprise. The exchange interface of our lungs relies on a mucous (liquid) so the gases can permeate the lung tissue. Expanding on that idea should certainly not rule out the possibility of liquid breathing.

The biggest drawback, besides the cost of a long term experiment and being dependent on a mechanical ventilator, is that these are very small case studies of success. Given the massive amounts of unknowns (particularly for long term studies), switching to purely liquid ventilation in an adult human could have unforeseen risks and would be considered dangerous by any ethics board (or insurance company)!

On a lighter note, here is a YouTube video of what Maddie, a biology podcaster from the BBC, had to say on the matter of living in total submersion. Lots of skin problems, risk of infection after a few days etc.

Again, a very interesting, albeit understudied, topic!