Does an algorithm exist for scheduling jobs on two processors?
The algorithms for $n$ processors will do the job just fine for the 2-processor case. In this context, $n$ does not refer to the number of tasks to be scheduled, which probably caused the confusion when you looked for algorithms.

Then, there is a great website for searching for scheduling problems, which you can access at http://www-desir.lip6.fr/~durrc/query/ -- You select the properties of your problem and will receive an overview of known results. Your case is called $P2 \mid r_j \mid C_\mathit{max}$ on that page. This is not quite your setting, as this assumes that the finishing times are the same for all tasks. But even in this slightly easier case, the web page gives you a reference for an NP-completeness proof for your problem:

Complexity of machine scheduling problems, Lenstra, J.K. and Rinnooy Kan, A.H.G. and Brucker, P., Ann. of Discrete Math., Vol. 1, p.343-362 1977.

So your algorithm will probably not yield a feasible schedule whenever there exists one.