You don't say which shell you're using. If it's a POSIX-compatible one such as Bash, then parameter expansion can do what you want:
Parameter Expansion
...
${parameter#word}
Remove Smallest Prefix Pattern.
The word is expanded to produce a pattern. The parameter expansion then results in parameter, with the smallest portion of the prefix matched by the pattern deleted.
In other words, you can write
$var="${var#*:}"
which will remove anything matching *: from $var (i.e. everything up to and including the first :). If you want to match up to the last :, then you could use ## in place of #.
This is all assuming that the part to remove does not contain : (true for IPv4 addresses, but not for IPv6 addresses)