Compress your links and save the compressed file to Dropbox
The previous replies are exhaustive and clear. I just suggest three workarounds, based on a single principle. The third looks more difficult but it is very efficient.
The idea is: compress your symlinks to a tar file, which Dropbox will not treat as a link, and save it to Dropbox. The third way is almost as direct as copying your links directly to your Dropbox -- actually much faster than copying several links one by one.
I have some twenty symbolic links I normally use in my Lubuntu installation, e.g. for quick access to bash scripts saved in Dropbox, to external hdds, to documents frequently used, etc. They are all in my home dir and in Desktop and I find it convenient to have a Dropbox copy of them to use for other installations or when I dist-upgrade (that is, when I reinstall after trying to dist-upgrade ;).
You make a directory called MyLinks (say, in Documents) and copy all your home symlinks to it; if you have symlinks in other directories, make subdirs in MyLinks and copy your symlinks to them. For example, if the links you want to save are in home and in Desktop, copy the home links to MyLinks, make a dir called Desktop in it and copy the Desktop links to it. Then open a Terminal, cd Documents (provided that this is the directory where MyLinks is) and make a tar file:
tar -cvf MyHomeLinks.tar MyHomeLinks
(Remember to add the ".tar" extension to the tarfile name, or Ubuntu Archive Manager would refuse to open it.)
If you do this, just be careful not to add the option -h (--dereference) to tar because it would precisely behave as Dropbox and compress the target files instead of the symlinks. (By the way, such a behavior would be welcome in most cases, esp. if one aims at portability, but it is not if one wants to save the symlinks as symlinks.)
The tar file would also include links that are temporarily broken, such as links to external hdds not currently mounted -- which is fine.
Save the tar file in your Dropbox. When you need the links back, copy the tar file to a directory in your file system (not within Dropbox), extract it (just right click and choose "Extract here") and copy your symbolic links back to the locations where you want them.
If the symlinks you want to save have a univocal prefix (such as "MySl"...) you can very quickly make your tarfile in this way (unnecessary spaces added to prevent mistakes):
find . -name "MySl*"|xargs tar cvf MyLinks.tar
The single dot after "find" means the current directory: open the Terminal in your home dir (as by default). You can surely add a path to the tar file and you can issue the command from another directory that is not home: replace the "." with the path to your home/user. No need to make work directories as MyLinks above. Restoring the symlinks is also quick: you copy the tarfile to your home, right click on it and "extract here".
However, before starting to prefix all the names of your symlinks, have a look at the next option.
3rd way (and best)
A very quick way for collecting all of your symlinks in a tar file is: open a Terminal in your home/user (which is default when you open a Terminal) and:
find . -type l|xargs tar -cvf AllLinks.tar
The drawback is, you get ALL of your links in it, including some mozilla or hplip symlinks you didn't even know you had. To avoid this and to copy all and only the symlinks from a specified list of directories without recursion, you write the list after "find" in the command above. Say you want to save the links from home/user and Desktop; you can write
find /home/<USER>/ /home/<USER>/Desktop/ -maxdepth 1 -type l|xargs tar -cvf MyLinks.tar
"-maxdepth 1" is to prevent find from recursing through home. You can add a path to tarfile if you wish. Alternatively:
find . Desktop -maxdepth 1 -type l|xargs tar -cvf MyLinks.tar
If you are lazy and you want to use this often, you can add it as an alias to .bashrc or to .bash_aliases in your home dir (.bash_aliases is a very convenient way for collecting your aliases and, why not, saving them to Dropbox...). Just create a new empty file called .bash_aliases in your home dir (that is, /home/user), open it with a text editor and write the following line in it:
alias save_symlinks='find . Desktop -maxdepth 1 -type l|xargs tar -cvf MyLinks.tar'
Replace the directories after "find" with the ones of your choice and "save_symlinks" with a command name of your choice.
To restore, open your File manager (if the path to the directories in the command above is not absolute, the tarfile must be in home/user), right click and extract.
If you do this, life is easier. All you have to do is:
SAVE: type save_symlinks in the Terminal, save the tarfile in your home to Dropbox;
RESTORE: copy the file from Dropbox to your home dir, right click on it and select "extract here".