I was just wondering what to do in this matchup. Once Annie has her stun up, harassing her doesn't seem feasible since I will lose due to her combo doing more damage.

Just be passive when she has her stun up, she won't hold it forever. When it's down then you can go for her. But overall annie seems to do pretty well against fizz (because of the reason you mentioned) or atleast better than a lot of people.

bait her stun out and juke with your E.
After 6, and she has stun loaded you KNOW she is going to go R Q W E combo.

You just know it.

So once she has that stun loaded, approach her, and the moment u get within her R ranged, you E E right on top of her. Hopefully behind her.
Activate W then Q through her, and poke her a couple times til u have to back out. U can escape through minions if you must. She will choke on her minions.
With her cd's she will only be able to land her W and Q on you. maybe 2 Q's, but with your W passive u will dot her enough, that a simple poke is enough to win the exchange and put her R on cd.

So I am basically hoping to get lucky to have her waste Tibbers? With her shield I don't do much damage, and dodging her skills that stun won't reset it, it only uses stun if it does damage to something. Every time she has a stun up and Tibbers it feels like I can't even try to go near her if she is smart.

Highly recommend you read this guide: http://www.solomid.net/guides.php?g=11920 written by Yako. Not only does this guide has comments on Fizz laning vs Annie, it has comments on almost every other possible mid opponents too, and a lot of commentary videos as example.

You will waster her stun, Plus with her shiled yes you wont do much, but that's why you just want to W Q her and get out.
Now her Tibber's will be on CD (after he dies) and you know her shiled will still be on a 20 cd when that's out.
When u tag her with W it's going to deal percent dmg based on her health missing over 3 second's. This won't feel like much.
But now that you've baited out her shield and tibber's, the next time she has her stun up you can try to juke it again.
If you jump on your spear while a turret shot is incoming, it will dissappear and you take nothing. This may be true for other projectiles, I don't know.
But even if you dont juke her stun, you can eat her burst and wail on her qith QW and E your way out before she get's too much dmg on you.. w.e you gotta do.

Use your E to dodge her stun, but when you dodge it, dont E away, make sure you E TOWARDS HER. Punish her for that stun. QE her and trade autos. PUNISHH HER