I think this depends on the pair of friends and the circumstances of the rivalry. Many friendships are broken up because of rivalry. Think about the entrance of a girl in between two boys or vice-versa. Lots of sets of friends agree not to date a person when both are interested in that same person. Every once in a while someone breaks the rule... so if that is the case, a friendship can be destroyed.
Some people are more sensitive than others. People who are overly sensitive might lose friends based on that character trait alone.
Gene seems rather sensitive. Look at the way he shows his sensitivity as a narrator. Especially during the trial of chapter 12, his thoughts do not mirror what is going on as he is hoping for it to go a different way.
Finny saves their relationship. He has every right to not be friends with Gene as Gene is jealous all the time. But he forgives Gene and the ability to forgive has to be there in order for a friendship to exist beyond every little bump of oa rivalry.
I think a true friendship can exist where rivalry is present. Most of our friendships are based on the proximities of school and work, where there is frequently some rivalry built in to the situation. Whether the friendship is harmed by the rivalry depends on the people, as it does in A Separate Peace.
Gene and Finney's relationship held the promise of a great friendship. Even though there was rivalry present because they were in the same year at the same school, they each had strengths that the other could have used. The reason that the friendship never became what it could have become, in my opinion, is because of Gene's insecurity and immaturity. He wanted to be Finney, or at least like Finney, and instead of offering the strengths he did have to Finney and accepting Finney's offered strengths, he tried to imitate Finney. Since he did not have the same qualities, he always fell short, and this caused him to be envious and destructive instead of being a good friend. Gene is not a terribly attractive character because he is not happy to be himself. This is, for many people, part of the process of growing up, learning to be yourself and to be happy about it!
In answer to your general question, I would argue that most friendships involve some type of rivalry. Whether those friendships last depends on what those involved choose to do with the rivalry. Sometimes it is used to motivate each friend to better himself, and other times it turns into jealousy which eventually destroys the friendship. Examples from history and literature demonstrate that when rivalry/competition exists in a relationship, usually one friend focuses too much on the rivalry, and the other friend seems focused more on the friendship and the success of his/her friend (i.e., David and Jonathan in the Bible, Gene and Finny, Amir and Hassan from The Kite Runner, etc.).
In A Separate Peace, Gene is the friend who thinks that Finny is competing against him (because he is insecure and cannot believe that someone could simply enjoy his company and friendship as Finny does). Finny wants what is best for Gene and really for all their friends. He doesn't care about breaking the swimming record for the school; he simply wants to enjoy life and see what his personal best is. Gene and Finny's relationship is on its way to being a true, lasting one when Finny dies in surgery; so it is realistic that the rivalry would have been set aside (on Gene's part) if Finny had survived.
Jonathan Swift, the British satirist, wrote in his "Verses on the Death of Dr. Swift" that friends always seek to be just a little better/higher than their friends. He states,
"I love my Friend as well as you, / But would not have him stop my View; / Then let him have the higher Post; / I ask but for an Inch [an inch higher than his friend] at most" (17-20).
Thus, you will have to decide if you agree with Swift that all friendships include some type of rivalry or envy, or if you think that his comments are not an accurate portrayal of human nature.