Altmetrics can showcase your broader impacts
Funding agencies like the NSF are increasingly asking researchers to document the "broader impacts" of their work. Altmetrics are a good way to do that, as they can help you find and explain how your research is being used by other researchers and the public.
A Research Consultation with a librarian can provide additional guidance.
Using altmetrics in your own promotion & tenure dossier
Some faculty are still unfamiliar with altmetrics, so do your homework before deciding whether or not to include altmetrics in your dossier. Ask around in your department with others who have recently gone up for P&T, and also your department chair, mentor, or anyone else familiar with the P&T process in your department and institution.
If you do choose to use altmetrics in your dossier, keep in mind that it's best to be selective with the metrics you plan to include. It's much more effective to include metrics that showcase the types of impact you're looking to document, rather than taking a "kitchen sink" approach (which might overwhelm your reviewers with numbers).
Altmetrics in promotion & tenure guidelines
Promotion & tenure preparation guidelines rarely include instructions on how to use impact metrics. Or, when they do, the guidelines usually only address citation metrics or, worse, recommend using journal impact factors.
These instructions often also lack guidance on how to make the metrics meaningful. For example, what does it mean if a tenure candidate says he received 5 citations for a paper published in 2013? Whether that's a good or bad number is often dependent upon the average citations that others in his field receive, and also the year the paper was published (as older papers tend to have more citations, by virtue of just being around longer).
There's an obvious need for clear instructions on how to use impact metrics in tenure & promotion dossiers. And there's also a need for guidelines to help dossier reviewers make sense of the numbers.
A small but growing number of universities include altmetrics in their tenure & promotion preparation guidelines. These include the University of Colorado Denver Medical School (PDF; page 84) and IUPUI (see: "The Guidelines for Preparing and Reviewing Promotion and Tenure Dossiers").
If you're interested in updating your university promotion & tenure guidelines to better document the use and interpretation of impact metrics, contact your faculty senate (or similar organization) to learn more about how that might work on your campus. You might also get in touch with your Vice Provost for Faculty & Academic Affairs (or similar campus office that oversees the writing of such guidelines).