Should I use present tense (spans) or past tense (spanned) in the following example?
"Dr X's service in providing data to the community {spans | spanned} across mission X (2003), mission Y (2008-present), and mission Z (2015-present)"
Do you mean to suggest that Dr. X is still providing data? In that case use the present perfect progressive, to express an action that began in the past and which is still ongoing:
He has been singing at church on Sundays for years.
In a situation like this, neither the simple present nor the simple past works:
He sings at church on Sundays for years (or since 2001)
is non-sensical, and
He sang at church since 2003 to the present
indicates that, as of today, his singing days are over.
But I think the verb choice you've made here is not the best. "Span" means to bridge or to cross, as in "an arch spanned the stream." So, span across is redundant, and the sentence is otherwise not idiomatic, to my ear at least. I would recast the entire thing:
Dr X has been providing data to the community since 2003, beginning with mission X and extending through mission Y (2008-present) and mission Z (2015-present)
The subject here is "the span of Dr X's service", which is a single thing. If that span includes the present day, then the span is current, ie in the present, and so you should say "spans".
Also, i think it's better to say "spans Mission X (2003)... etc" rather than "spans across": both "span" and "across" are to do with covering an area, so to use both is redundant.