I have the same issue and so do many people who are currently using Poloniex. The two-factor system still works, it is just lagged on their end. They have have had a major influx of new users and their current servers are not up to handling that work load. Others say they aren't adding servers fast enough and that it is unacceptable, but I look at it through a security lens. They are ensuring that the security of new servers is good enough since attacks are common for trading networks.

How to login right now - write down your two-factor code and wait until the next cycle is finished. Then try using the code you wrote down. This means you are putting in the two-factor code from 2 previous code cycles. I know this seems dumb but it works since their end is lagging.
e.g. it says 123456 - write that down. The next code is 234567, wait until this code has timed out and you see the third code 345678. When you see 345678 you type in 123456 and it should work.

Good luck!