What is the origin of “call a spade a spade” and does it have racial connotations?

It goes back to at least 1542, so I can confidently say there are no racist origins in the expression.

Apparently it first appeared in English in Nicolas Udall's collection of Erasmus's aphorisms - translated in 1542, but ultimately deriving from Plutarch's Moralia in the first century AD.

It's really just an observation that forthright honest people use straightforward words. I doubt the fact that "a spade" happens to be the common example has any special significance, though that obviously wouldn't have worked for Shakespeare in the related rose by any other name.

This is answered fairly clearly by the etymonline entry for spade:

spade (1) "tool for digging," O.E. spadu, from P.Gmc. *spadon [...] from PIE *spe- "long, flat piece of wood" (cf. Gk. spathe "wooden blade, paddle," [...] To call a spade a spade "use blunt language" (1540s) translates a Greek proverb (known to Lucian), ten skaphen skaphen legein "to call a bowl a bowl," but Erasmus mistook Gk. skaphe "trough, bowl" for a derivative of the stem of skaptein "to dig," and the mistake has stuck.

The item before that in etymonline mentions

spade (2) ... Derogatory meaning "black person" is 1928, from the color of the playing card symbol,

an unrelated meaning.

The origin of the phrase doesn't have any racial connotations (as jwpat7 and FumbleFingers showed), but you should be careful how you use it because of the derogatory meaning of the word spade.

In some situations it would be clear that only the original meaning was intended, but in other situations it might be misinterpreted as wordplay on both expressions.