How do I make LaTeX push long citations to a new line?

Without a minimal example (i.e., code that will show the problematic behaviour), one can only speculate. However, I've stumbled over this problem in the past, and it may (!) help if you add the following to your preamble:

\usepackage{breakcites}

See the breakcites page at CTAN for details. If it doesn't help, you'll have to provide a minimal example that (above all) shows your document class and your bibliography style.

The "general solution" for overfull boxes in paragraph mode runs as follows:

1. Rewrite your text. Seriously. In some cases, this can be the only way to achieve "perfect" typesetting.

2. Enable the microtype package. Surprisingly enough, this package reduces many occurrences of overfull lines.

3. Wrap the paragraph with \begin{sloppypar}...\end{sloppypar}. This relaxes LaTeX's rules for how much interword space is acceptable; you'll get less "grey" paragraph, but at least it will stay within its margins.

4. Finally, if this happens a lot, you can enable \sloppy for the entire document. With this mode in effect, well-typeset paragraphs will (almost almost always) remain just as well typeset, but "bad" paragraphs such as you're talking about will behave as if they were wrapped with sloppypar. I tend not to recommend this route as it can make you lazier about good typesetting—but depending on your document this is not necessarily a bad thing.

In order to split citation you must add both these packages, otherwise the citation will not properly split.

\usepackage[breaklinks=true]{hyperref}
\usepackage{breakcites}