F430 key remote only works when cover is off

Battery contact issues are a common problem. I don't know if I had quite the same problem as you, but I solved mine by putting a few folds of paper under the battery. It seemed to reposition the battery and give it better contact.

 

On the 360 there was an upgrade you could do by moving the antenna to improve distance. I did it on my car and it helped a lot. Could be a similar situation on the 430 as well.

Ray

 

That "upgrade" is explained here, tip at the end Ferrari Fobs 101

 

Sometimes the sprung metal tabs that press against and make contact with the battery lost their springiness and don't provide enough pressure against the battery to make contact.
You can fix this by taking the battery out, getting a very small flat screwdriver, and bending the tabs out slightly.
They should press tightly against the battery when you put it back in. Be careful not to overdo it. Only pry/bend in a direction that wont break the circuit board or plastic mounts.

Sometimes there is oxidation or corrosion on the tabs that should be cleaned. A #2 pencil eraser works well for this.
Sometimes stress or vibration cause cold solder joints on the circuit board. This requires a solder reflow.

 

Just after I posted my response, I decided to replace the batteries in my keyfobs and one of them didn't respond Indeed, it was a soldered joint problem on one of the battery contacts.

It's difficult to get enough heat into the joint because the battery contact is conducting the heat away. You need a soldering iron big enough to heat up the battery contact, but not big enough to damage the circuit board.