I have a player looking at taking the Warlock Repelling Blast invocation, which allows a character to push a target 10 feet away from the caster. The player is curious about if this would deal any damage or have any other effects if a target is pushed into another, or a solid object like a wall.

My gut instinct is no, since the 'push' effect doesn't seem like something that would generate enough force to cause damage, but does anyone have any thoughts?

I would also say 'no'.  It's much like the action listed under "Shoving a Creature" in the Combat chapter, which also does not say it does any sort of damage, or have any potential for damage, from pushing a creature.

Now, if you're say, pushing a creature into a big spiky wall, I would say that has the potential to do additional damage based on the exact situation and at the discretion of the DM.

I do not know if there is some other similar enough spell or ability that could provide mechanical insight into pushing one creature into another.  My instinct tells me that it should be allowed, since battlefield control through moving yourself and opponents around in specific ways has been a tenet of D&D since at least 3rd edition.  I'd say if the character is pushing Creature A into Creature B, Creature B probably needs to make some sort of ability check to avoid issues.  Perhaps Dex to avoid any consequences, meaning Creature A passes through the square unhindered, or Str to stop Creature A from being pushed through, with Creature B getting to choose which check it wants to try and make.  If it fails the check maybe Creature B is either pushed along as well, or is knocked prone.

It still leaves it as an option, but it doesn't make it an all powerful tactic.

Pushing into a pool of acid: Damage.
Pushing into a regular wall: Just mild discomfort.
Pushing into a spellcaster: Spellcaster gets distracted.

I'd say that's really about it.

This is always an interesting debate. I'm not certain how this works in 5 but in previous versions if you push one person into another then you end up pushing both. I have seen a fighter team bullrush back an entire team of goblins off a cliff before. I have also seen house rules where bullrush let's you crush a person into a wall for 1d4+Str per 10ft of extra movement. If you have ever been checked into the wall in hockey you are familiar with this.

Now here is the question. Can the same rule be applied fairly to magical effects that "push" a person. I would believe not simply due to the gentle nature of a pushing magic vs the violent nature of a bullrush.

Gentle nature?

http://www.d20srd.org/srd/spells/forcefulHand.htm

Pushed down a flight of stairs, over a ledge, in to a pit, on to your comrade (who has a sword, knife, or other sharp object which may stab in to the one pushed), pushed on to a Grease spell area (slides are no fun backwards, especially if your not expecting them), especially if it is one of the flammable Grease effects (pushed in to Grease zone, then someone sets zone on fire). Pushed in to a fire, in to path of big monster or berserker.

Push can be a very dangerous spell if used right. It's all about when, where, and what you can shove the target in to for best effect. Also (and this is one I have always thought should apply) the one pushed Should have to roll a Dex save to see it they fall down. That could have other effects to, fall wrong and you can break something (I broke my wrist cause I fell wrong once).

Also nothing in the Push spell says it has to be a Living target. Push a heavy object off a high spot and you could squash someone. Or use it to cause a catapult effect, at the correct angle of pressure you could send objects flying through the air. Pushing someone back 10ft is a good amount of pressure after all.
Try it sometime, you will see just how much force it takes to move something 10ft just by pushing on it.

In the spells description it specifies,

There for this is a bullrush and not a pushing magic.

This is a very well made point. I think creativity like this should always be rewarded in game. But let's not forget you have a level 10 character with 50-100 hp. Lets see a normal human take that kind of punishment. When making a gm call about how much force that is one must always remember the setting in which it takes place as this will greatly affect things.