It depends on how sophisticated you need this. If all you need is to change the 01 to a 02, you can use something like
sed -i 's/01/02/' file
or
perl -i -pe 's/01/02/' file
Or, to be on the safe side, do it only if the 01 is at the end of the line:
sed -i 's/01$/02/' file
perl -i -pe 's/01$/02/' file
Both of the above solutions will modify the original file because of the -i flag.
If you need to be able to take actual dates into account and, for example, increment 2014-02-28 to 2014-03-01, you need to either use a full-fledged programming language that can parse dates or play around with the date command:
$ while IFS='=' read -r text date; do
echo "$text=$(date -d "$date + 1 day" +%F)";
done < file > new_file
The above will split input lines on =, saving the part before the = as $text and the rest as $date. Then, the $date is fed to the date command which can do fancy date manipulations like add a day and then print it out in the format you want (in this case, the format is %F which means Year-Month-Day). Note that this assumes GNU date.
The echo command prints the contents of $text and then the date incremented by one day. This will fail if you have more than one = on any line of your file.