Will I get banned for logging in from different IP addresses?

[Kraggy]

No. You cannot.

Incorrect as a blanket statement, as LilPhist posted.

While such a 'ban' is only temporary (technically it's a 'lockout') in the past certainly SE have used geo-location to try to detect account thefts; one of my SE Accounts suffered from this when playing V1 when I switched ISPs. SE also hinted though never explicitly stated that there was a threshold on the number of such locks before they took it more seriously.

That said, with all the VPNing people have been doing to avoid P2P throttling it seems clear SE have (for the time being at least) stopped using geo-location. Certainly I've used a US-based VPN (like many) and SE didn't say a word, while Blizzard locked me out as expected.