Over finite fields, the Tate conjecture by itself doesn't imply the standard conjectures, but together with the Hodge conjecture for CM abelian varieties it does (Milne 2002, 2009).
In a little more detail: Milne showed that the Hodge conjecture for CM abelian varieties implies the Tate conjecture for all abelian varieties over the algebraically closure k of a finite field in a 1999 paper (Compositio), and that it implies the Hodge standard conjecture for AVs/k in a 2002 Annals paper. If you assume the Tate conjecture, then the category of motives over k is generated by abelian varieties (mentioned in Milne 1994, Seattle I). In terms of motives, the Hodge standard conjecture says that the pairings defined by ample line bundles are positive for a polarization on the Tannakian category of motives.