Approval affects Production, Influence, Pop Growth, and Resistance. It's very important.

Approval only affects organic population growth, resistance and influence and has been this way since Beta 5.  Previous to that it did affect production.  

I don't like it and if it stays that way at release I plan on modding it out of the game.  At the moment I generally ignore morale with Synthetic races but try to keep it at a reasonable rate for Organics just to keep the growth bonus.

Not really.  You'd have planets not growing as you build/upgrade farms and not generating as much influence as they should, therefore they risk flipping to another nation's side.

I say you can ignore approval buildings very early in the game, while you're rushing for colonies and very late in the game, when planets are fille to the rim with people.

There is no tooltip that says approval affects production...however brad did say it does in one stream.

i have seen some prod numbers I couldn't explain so this might be it

It appears that it does indeed still affect production.  Not only that, but research as well.

I have a simple Before and After below:

I ran across an anomaly that finished my research on a +4 Morale tech, giving me a perfect Compare and Contrast.  Thus the only difference in the two screenshots is approval.

It affects income as well.  Curious, I decided to go back and raise my planetary slider to 33/33/33 and then run the experiment again. 

Before and After from planetary tooltip:

Before on the Planetary Wheel:

After on the Planetary Wheel:

Makes me think it is affecting Raw Production (I think that's the one at least).  There's been quite a few problems with that and tool tips.  If it's not Raw Production, it's whatever the term is that described what goes into the Planetary Wheel.

EDIT:::  And, no, it's not because I was running an Economic Stimulus.  I ran the experiment yet again and this time looked at a planet that was producing improvements as normal.  Its income went up as well (no screenshots this time).  So approval is absolutely affecting things. Not sure what the percentage is though.  I want to say 25% since that's what everything else is, but the math doesn't work out right.  Though since Raw Production (IIRC) is on a curve, that might be messing things up.  I'm sure someone with better math chops than I might be able to reverse engineer it, especially if they have the formula for pop to raw production on hand.

Oh, I absolutely agree that approval should affect production.  I was irked that it apparently hadn't as of Beta 5.  Now that it is shown to change things, I feel better.  Just wish I knew exactly what was going on.

Approval only affects production when you are at 100. When at 100 I believe you get a 50% boost or 100% boost to everything. However below that all the way to zero you will not take a hit to production. 

Thus its in you favor to get as many planets at 100 approval as you can. This is one major reason why I play with very few habitable planets in game. On very large maps with common planets you will end up with over a 100 colonies and at a 1% disapproval per planet you will sink happiness faster than talking about Hillary for Pres......

On that note I found that one approval building and a starbase with a rec center will generally do the job for that solar system. Also approval relics are a must have actually as their bonuses 'should' stack....