The first few unstable homotopy groups of the unitary groups $U(n)$ were calculated by Borel-Hirzebruch, Toda, and Kervaire, and they are all torsion. There is a paper by Matsunaga (details below) in which the p-primary parts of the next few homotopy groups are calculated. This paper claims to be giving a complete description of these homotopy groups, but I don't see any explanation of why there are no free abelian factors.

My question is: Why $\pi_{2n+i} (U(n))$ is a torsion group for $i=3,4,5$?

More generally, is it know whether or not $\pi_* (U(n))$ is torsion for all $*>2n-1$?

Matsunaga's paper is:

MR182010 Matsunaga, Hiromichi Unstable homotopy groups of unitary groups (odd primary components). Osaka J. Math. 1 1964 no. 1, 15–24.

It's freely available here: https://www.jstage.jst.go.jp/article/kyushumfs/15/1/15_1_72/_article

There's actually an errata list for the article that's not contained in the freely available .pdf. It just corrects various notational errors (though not the obvious notational error in the main theorem, in which part b should refer to the 3-primary summand).