Interesting question, have never thought about it. Not sure I've ever seen it happen. I guess as said above by RedRabbit Kubica's accident shows it can happen, but also the downsides.
The NBA is a really good example of teams having huge amounts of money invested in players, their franchise depends on it but every year in the off season there are world championships or Olympics, where you have to weigh up the risks vs the rewards because for some playing for their country is the ultimate honour and for others.. it's a chance to rest and recharge for the next season. There are always the occasional injury but overall it seems a personal decision. A few months ago Paul George snapped his leg bone in two while playing for Team USA and now his Indiana Pacers suffer because of it. Some talk about including contracts that forbid it and others just let the player be happy making their own choice. It can lead to resentment otherwise. They are sportsmen and it's their job to play the sport they love. There is a conflict of interest in terms of doing what they love vs risking injury. In football, I don't follow it closely but it seems a given that they will always want to play for their country. One of the rare times where they will always want to play for the country but I guess the World Cup is the center stage?
In F1 it seems very rare and the seasons are structured differently. There is no higher stage than F1, and since that is what they do 9 months out of every year, the other 3 months seem best just to rest and prepare for the next one. They do other sports like cycling or triathalons but it's rare to see them appear in other racing series. In racing it's a bit different because different racing series are almost like different sports, with a different approach so it's not as easy to transition between the two. It'd be great to see F1 drivers appear in NASCAR or Indycar during one off events (between F1 races) but it's not a simple thing. The best you can hope for I guess is those days where they just do a test session to try it out.. for media promotion purposes. It was pretty cool to see Lewis try out a NASCAR at Watkins Glen.
Pretty sure the ultimate F1 driver fans dream is to see them all race on a level playing field.. but then again, F1 heads more and more towards a spec series anyway.