Ah, trust, that fickle thing...
Tor provides anonymity for the download part. A download is: to obtain a sequence of bytes. What you do with these bytes is then completely up to you.
Some sequences of bytes encode executable instructions that a computer will be eager to run. Executable files, scripts... fall in that category. If the file you download contains instruction, and these instructions have been designed to be hostile to you and your anonymity, and you execute them nonetheless, well, then you get what you asked for. The warning popup displayed by Tor is a kind of disclaimer: it reminds you that the magic of Tor stops at the downloading, but does not guarantee that the file you obtained is not full of nastiness.
Now for Word documents. Theoretically, a Word file contains the description of a written document, possibly with pictures; but, in practice, a Word document can embed just about anything, including executable applications. Word also supports a complex system of macros, which are, by any reasonable definition, a programming language. Thus, "opening" a Word document is quite akin to running a script. And, indeed, macro virus do exist.
Even with macros disabled, some nifty attacks against anonymity can be performed with Word files. For instance, Word documents can be signed. Word will want to verify this signature, which means first validating some X.509 certificates, which in turn may make your computer download some intermediate CA certificates and/or CRL by following URL found in the certificates themselves. As such, a Word document which you merely open may imply network activity to target names that are embedded in the document (well, in certificates which are embedded in the document). The nice part is that these accesses will be performed by some system components which may completely disregard your browser configuration -- thus happening outside of the Tor umbrella. Goodbye anonymity !
So don't open potentially hostile Word documents. However, if you trust the file, then there is no problem, yes ? At least as long as you can be sure that the file you got is really the one you believe it is... Amusingly enough, digital signatures can help you there, but the mere act of verifying the signature can entail activity which makes you totally non-anonymous, as explained above.
(The same applies to PDF, Excel...)