Many algorithmic graph problems can be solved in polynomial time both on unweighted and weighted graphs. Some examples are shortest path, min spanning tree, longest path (in directed acyclic graphs), max flow, min cut, max matching, optimum arborescence, certain densest subgraph problems, max disjoint directed cuts, max clique in certain graph classes, max independent set in certain graph classes, various max disjoint path problems, etc.
There are, however, some (although probably significantly fewer) problems that are solvable in polynomial time in the unweighted case, but become hard (or have open status) in the weighted case. Here are two examples:
Given the $n$-vertex complete graph, and an integer $k\geq 1$, find a spanning $k$-connected subgraph with the minimum possible number of edges. This is solvable in polynomial time, using a theorem of F. Harary, which tells the structure of the optimal graphs. On the other hand, if the edges are weighted, then finding the minimum weight $k$-connected spanning subgraph is $NP$-hard.
A recent (Dec 2012) paper of S. Chechik, M.P. Johnson, M. Parter, and D. Peleg (see http://arxiv.org/pdf/1212.6176v1.pdf) considers, among other things, a path problem they call Minimum Exposure Path. Here one looks for a path between two specified nodes, such that the number of nodes on the path, plus the number of nodes that have a neighbor on the path is minimum. They prove that in bounded degree graphs this can be solved in polynomial time for the unweighted case, but becomes $NP$-hard in the weighted case, even with degree bound 4. (Note: The reference was found as an answer to the question What is the complexity of this path problem?)
What are some other interesting problems of this nature, that is, when switching to the weighted version causes a "complexity jump?"