I suspect you're thinking that pulling a string wrapped round a cylinder would compress it in the same way that pulling on a string wrapped round a coke can would crush it. If so, this won't happen because the tension in the string can't produce any force out of the 2+1D manifold that it lives in.
However you are quite correct that the tension in the string will contribute to the stress-energy tensor, and therefore act as a source of gravity. I have to confess I don't know what the effect would be in 2+1D, however in 3+1D the effect would be similar to a cosmic string. Assuming the mass of the string is negligable, so the tension is the only contribution to the stress-energy tensor, the spacetime around the string is flat but has an angular deficit. That means you don't feel any gravitational field from the string, but if you travel in a circle around the string you'd find you had rotated by less than 360º.
I suppose the equivalent to your 1D object in the 2+1D universe would be a 2D object in our 3+1D universe i.e. instead of a stretched string you'd have a stretched membrane. An example of this would be a domain wall. This does produce a gravitational field, and in fact the field is repulsive i.e. you would be accelerated away from the wall in a direction normal to it. There is a description of the field produced in this book.