Well, that's odd.
 

I recently upgraded my bike from a triple to a double without changing the bottom bracket, which was sized for a triple.

Now, if I am in the largest cog in the back and in the front, when I pedal backward the chain will actually shift down from the top.

I'm guessing that isn't normal and my bottom bracket is too wide. The drive train is completely silent when pushing forward, however.

Do I need to change my bottom bracket?

One of my bikes will do that. It is 53/39 front and 9x/34/11 rear. I just avoid pedalling backwards in the 53/34 combination. That 53/34 combination is cross-chained and, in theory, should be avoided anyway. The problem is worse if the bike has a short wheelbase. On my bike, there isn't room to bring the crankset in. I suspect you should just learn to live with this and avoid the situation that derails the chain. how is chain alignment in more appropriate combinations?

Don in Austion

Stick a yardstick between your front chainrings. Notice where it bisects the rear cassette. If it hits near the middle, you're good. If it hits close to the smallest rear cogs, your BB is probably a little too wide.

If the bike never did this when on the middle of the triple, then you probably changed the chainline when you upgraded. Or to look at it another way, had you "upgraded" by simply removing the granny you wouldn't be here.

So it's fairly safe to conclude that the new cranks had different spindle offset that the ones you removed, and both chainrings are more outboard than your old middle and outer. So, even without measuring, you need a shorter spindle.

You can get a sense of how much shorter by looking at your chainring/chainstay clearance which is typically only a few millimeters. Or you can do a proper chainline measurement to determine exactly where you want the crank to end up.

I actually changed the crankset, don't worry. I'd figured that a double would less spaced outward, but I'm guessing I was wrong. Though, in the big big combination on the triple this didn't happen.

It's not double vs. triple that made the difference, it's generational.

Over the years spindles have gotten shorter, with more offset built into the crank arms. This improves rigidity, and olse gives some extra ankle clearance by way of the curved crank arms. But it means that a newer crank on an older spindle will be too far out, and vice versa.

If you told us what parts you're using (brand/model) we might be able to actually provide some useful information about your specific problem. Makes sense, doesn't it?

It's not an outdated bike--the triple is a 2012 model from FSA, and a 2013 FSA double replaced it. The bottom bracket is a 127mm UN26, I believe.

FSA has a customer service number. You can ask them for the correct spindle length for your specific crankset. There's no reason to run an incorrect chainline.

127 is quite long for a modern crank. i am guessing you need a 110 or around that range. i am talking in general here. consult the owners manual for what you need for proper chainline.