Machine learning techniques for estimating users' age based on Facebook sites they like
One thing to start off with would be k-NN.  The idea here is that you have a user/item matrix and for some of the users you have a reported age.  The age for a person in the user item matrix might be well determined by something like the mean or median age of some nearest neighbors in the item space.

So you have each user expressed as a vector in item space, find the k nearest neighbors and assign the vector in question some summary stat of the nearest neighbor ages.  You can choose k on a distance cutoff or more realistically by iteratively assigning ages to a train hold out and choosing the k that minimizes the error in that assignment.

If the dimensionality is a problem you can easily perform reduction in this setup by single value decomposition choosing the m vectors that capture the most variance across the group.

In all cases since each feature is binary it seems that cosine similarity would be your go to distance metric.

I need to think a bit more about other approaches (regression, rf, etc...) given the narrow focus of your feature space (all variants of the same action, liking) I think the user/item approach might be the best.

One note of caution, if the ages you have for train are self reported you might need to correct some of them.  People on facebook tend to report ages in the decade they were born.  Plot a histogram of the birth dates (derived from ages) and see if you have spikes at decades like 70s, 80s, 90s.