If you are using a filesystem type that stores timestamps in UTC or equivalent (like Unix timestamps for many Unix-like filesystems), then daylight savings time should not be a factor, as long as your system(s) has correct UTC time.
Unix timestamps are always internally processed in UTC, and the conversion for the local timezone (and DST if applicable) is applied only when displaying them.
When mounting FAT32 filesystems, you can use the
time_offset=<number_of_minutes> mount option to specify the timezone used by that particular filesystem, if it is different from the system default timezone. (For example, if that filesystem comes from a portable device that has no clue about timezones or UTC.)
As far as I know, any version of
tar I'm aware of supports checksums only within the structure of the
.tar archive; they will not support adding/updating files into the archive based on the result of checksum comparision. If you need to check for file modification using checksums, my first thought would be to look at
rsync and its many options. But it just copies files from A to B; it does not create nor maintain archive files.