If you are interested in numerical (approximative) solution, one of many methods that I would recommend (and I use it) is an "immersed interface" method (or several other similar methods with similar names like ghost fluid methods etc). In such a case you need to specify boundary conditions on your (moving and fixed parts of) boundary.
The idea is that your moving boundary is represented as a zero set of some function (the level set function, usually a signed distance function). To solve the Laplace equation numerically, you can use very simple finite difference method on structured (quadrangle) grid that involves always your moving boundary. As you want to solve it only for grid nodes where your level set function is (e.g.) negative and not there where it is positive, you have to apply some "extension" (extrapolation) for the nodes next to your boundary. If your boundary conditions on the moving boundary are Dirichlet type (the values of solution are prescribed) then this extension is straightforward.
As I am not sure if you are interested in numerical solution, I stop my description here. As a literature I would recommend:
F. Gibou, R.P. Fedkiw, L.T. Cheng, M. Kang, A second-order-accurate
symmetric discretization of the Poisson equation on irregular domains,
J. Comput. Phys. 176 (2002) 205-227
that is available on the net and you should look only for the stuff dealing with Poisson equation.
I was using such method for moving groundwater table:
P. Frolkovič: Application of level set method for groundwater flow with moving boundary. Advances in Water Resources, Volume 47, October 2012, Pages 56–66
Fancy animation ;-) below