Fort and Hedlund determined the minimum size of a $(v,3,2)$-covering design: [Minimal coverings of pairs by triples, Pacific J. Math. 8(4), 709-719, 1958.][1] 

The [La Jolla Covering repository][2] is a good source for covering designs. According to [this table][3] the smallest open case with $t=2$ is $(v,k,t)=(23,6,2)$ where the minimum size is either 20 or 21.
 

  [1]: http://projecteuclid.org/euclid.pjm/1103039697
  [2]: https://www.ccrwest.org/cover.html
  [3]: http://www.ccrwest.org/cover/low_tab.html