The short answer is: kinda, and it won’t be perfect.
Here’s the deal. DEVONthink and Finder understand each others’ tags, and tags are a property of a given file.
What exactly this means depends a bit on some aspects of your workflow.
Finder->DEVONthink
If DocumentX is in Finder and you apply a Finder tag TagA, then TagA is a property of DocumentX. In Finder, if you click to see all the files with TagA, you’ll see DocumentX.
Now, if you IMPORT that file into DEVONthink, that file and its tag are now part of whichever database you imported it into. So in that database, if you click on the Tags group, you’ll now see TagA has been created (if it didn’t already exist), and that DocumentX is listed as having been tagged this TagA.
DEVONthink->Finder
Exactly the same as above except reversed. DocumentY is in DEVONthink, you tag it with TagB, when you export that file to a location in Finder (or drag it out of DEVONthink into Finder) then the document remains tagged, and the tag is created in Finder (if it didn’t already exist).
DEVONthink->DEVONthink
If you move a document between databases, then the same principles apply. The tag remains a property of the document and is created in the receiving database if it doens’t already exist.
Indexed Files
Things get a little finicky if you are INDEXING into DEVONthink files that lie in some location in Finder. In this case, all of the above principles apply, but it can be a bit flakey. Indexing is pretty solid but does require a little bit of care and is not bulletproof.
With indexed files, lets say a text file, if you edit the document in DEVONthink, the change is visible in Finder, of course. The same applies in the other direction. If you use Finder to open the text file in BBEdit and make a change, that change is visible in DEVONthink.
Similar to editing the contents of an indexed text file, since tags are a property of the document, then changes made in DEVONthink should be pushed to Finder and vice versa. In practice I find it’s a bit less seamless with tags.
If you have a file that is already indexed in DEVONthink which you subsequently tag in DEVONthink, or tag in Finder, you need to force DEVONthink to “Update indexed items” to get the tags for a given document to reconcile between DEVONthink and Finder.
In all of the above cases, deleting a tag (NOT removing the tag from a file, but deleting the actual tag) is not synced – that is, if you delete a tag in DEVONthink that had been applied to 50 indexed files, the tag will be deleted and removed from those files. Once indexing updates everything, the tags will also be removed from those files in Finder however the tag will still exist in Finder (it just won’t be applied to those 50 files from which it was just removed in DEVONthink). This means that deleting an actual tag in one location will not delete it from all locations.
This kinda covers the basic possible scenarios. What exactly is your scenario? Perhaps if you tel us whether you are indexing or importing, and other details, we can offer you a more specific an detailed set of suggestions.