First of all, what type of geometry are we dealing with here? Is the vase polygons or nurbs? I know the star shape is a nurbs curve.

The easiest way to do it would be with polygons, and I'd even go so far as to say that it'd be almost impossible with nurbs. Almost.

If your vase is polygons. Good. If not, you can convert it. I have a Nurbs > Polygons conversion vid at my personal site if needed.

Take your star shape, duplicate it and move it to the side. Select them both and Surfaces > Loft > Options. In the options, you can use polygon outputs. So, you'll use 2 nurbs curves to output a polygonal "cylinder" that is in the shape of a star. An elongated Star, so to speak. You'll need to cap the top and bottom by using the Append to Poygon Tool (under Polygon menu). The way Append works, is you click on the tool, and then click an edge along the upper lip of the star cylinder. You should see small purple arrows appear. You may need to zoom in to see them clearly.

Click on the edges one at a time, in the direction the small purple arrows point. This will fill that area with a pink surface. When you're all the way around, click enter. This has created a face to fill that hole. Do the same for the opposite side.

So, now you've got a polygonal vase structure, and an enclosed, elongated star shaped "cylinder."

First thing you should be aware of is that Maya's booleans are buggy. They have been improved in their ability to work, but, at least in my case, occasionally, the 3 available boolean functions seem to do one of the other actions.

For example, Difference may sometimes act as Intersection and vice versa. That may just be me, but I'm not sure.

In any case, insert the elongated star cylinder into the polygonal vase where you want the hole to be. Select the vase, and shift select the star shape.

Go to Polygons > Booleans > and pick one. If it does not do the action you want, undo and try another one. Do so until you get the star shape cut into the vase. If all 3 do not want to work, try selecting them in the reverse order... selecting the star shape first, and shift selecting the vase.

If none of that works, as it does happen occasionally, you'll have to manually cut a star into the vase. With the star shape still inserted where you want the star located, you can use the Split Polygon Tool to outline the cylindrical star and make a star cut into the vase.

Good luck.