In Romeo and Juliet, does Shakespeare seem to consider a self-destructive tendency inextricably connected with love, or is it a separate issue?

That's an interesting question. The two characters in the play who fall in romantic love do commit suicide, so it could be easily to conclude that Shakespeare is pointing the finger at romantic love and saying it is inextricably connected with self-destructive tendencies.
However, I would argue this is not the case. As Shakespeare points out in the play's prologue, it is the feud between the Capulets and the Montagues that makes these two particular lovers fated to self-destruction. This idea is reinforced throughout the play. For example, Juliet bemoans the fact that she has fallen most in love with what she is most supposed to hate, and she tells Romeo during the balcony scene that the problems they face lie in his being a Montague: a rose by another name would smell as sweet, but, unfortunately, she knows that her family will not get beyond the name. Throughout the play, the two lovers are constrained and forced into secrecy by the feud.
We can also point to the impulsive nature of the two lovers themselves—Romeo in particular, though Juliet shares his feelings. They have to marry immediately and have it all now. This leads them into the final fix that results in their suicides.