I wrote a couple of utilities that let you use shapes to draw polygons in Tableau in two ways (one using a supported approach in Tableau, the other is a bit of a hack). Both of those use an open source component which supports a whole lot of different file formats, including MapInfo. I think I even tested both of my tools with MapInfo format.
From Mapinfo you can select the region you want and then go to the Table menu and export as a .MIF file. You can then right click on the file an open it in MS Excel and grab the Lat and Long. Try it with the USA.Tab file from Mapinfo and pick a large state so you can try it out.
You have a couple of options now with Tableau, paid for; and open source.
If you are analyzing your shape files, then Richard's tool is excellent.
You can also use a mix of Quantum GIS, Postgres SQL with an instance of PostGIS (as an alternative to ESRI) to import and analyse the shapefiles.
If you want the shapes as a background image, then you can install Geoserver (or you could ask here) and host these. Geoserver can also publish PostGIS tables. The shapefiles would need to be styled...I use AtlasStyler for this.
Siebe Bosch has successfully used Map Window, and has some code that does a similar job to Richards.
I haven't attempted Hector's solution, it maybe more elegant that the others?
The other service is to use urbanmapping.com, Tableau's third party mapping solution, they are able to host the shapefiles for you, but this is a paid for service.