I suspect that xargs is your friend in this instance.
# cat file | xargs -IREPLACE rsync -v REPLACE server:/path
xargs will take data from stdin and add it to the commandline one parameter per input line until it fills the command line length, and then runs a separate command until all the input lines have been used up. This will mean that rsync could well run more than once. This shouldn't be a problem in this case, but could be with other commands.
Usually xargs just sticks the arguments on the end of the line, but with rsync you'd want to have the destination last, so we use the -I option to replace a string with the arguments. (This is the bit that's untested). I'm also a little unsure what rsync will do with the paths, so please test this carefully before using it in anger.
If you do have a problem with xargs, you could use something like:
for i in $(cat file); do rsync -v $i server:/path; done
If your file is larger, you probably want to use a while/read loop instead. This will be much slower as it will do a single file at a time.