Dissonant Whispers kind of does this already
Having recently played a bard in an extended campaign, I got a lot of mileage out of Dissonant Whispers, which uniquely forces movement that provokes opportunity attacks:
On a failed save, it takes 3d6 psychic damage and must immediately use its reaction, if available, to move as far as its speed allows away from you.
It will allow for crazy combos
Most of the time, the damage output won't be increased that much, because most creatures only have one reaction. However, it's possible to set up very damaging combos against single targets.
For example, imagine an enemy surrounded by a paladin, a warlock, and an eldritch knight. On the warlock's turn, he casts eldritch blast and hits, allowing him to push his target 10ft without a save. The paladin and eldritch knight both get opportunity attacks: the paladin gets to smite, and the eldritch knight, with war caster, gets to cast a spell. Moreover, because the warlock can shoot multiple eldritch blasts, this combo could theoretically be triggered multiple times in one turn if there are enough friendly creatures to make OAs. You can easily see how powerful such a combo can be.
Of course, Dissonant Whispers means that these combos are possible in vanilla 5e, but your tweak allows classes like warlocks to do it for free and without a save. Whether or not that breaks your game will depend a lot on how your PCs and monsters are built, but it opens the possibility of very powerful single-target combo attacks.
I realize that these combos are pretty much the intent of this houserule, but my point is that a party that builds around this houserule will become far more powerful than a party that doesn't.