Please open your console.log and search for:
If you see this error, this sticky applies to you.
Checksum for file :/update/game/resources0.s2z/checksums does not match!
MODIFIED RESOURCES0.S2Z DETECTED for file :/update/game/resources0.s2z/checksums
There seems to be a problem with one of the files on one of the Content Distribution Network (CDN) nodes. One possible workaround is to use a proxy program such as Hotspot Shield to try and grab the files from a different node. You must do this from a clean HoN install (reinstall the 2.0.2 installer and don't run it until the proxy is on). I don't have access to try and fix this stuff and cannot build the 2.0.3 installer, unfortunately.
Edit1: That doesn't seem to get around the problem, unfortunately. Right now there's nothing more I can do until I can get someone to check the CDN, which will most likely be Monday.
Edit2: Someone had success updating using a pre 2.0.0 installer. Unfortunately they seem to be removed from the download servers, but I'll try to get something working.
Edit3: The 2.0.3 installer is out, now.