Integral versus real (universal) characteristic classes

If $X$ is a space of finite type (meaning that the homology groups $H_i(X)$ are all finitely generated, a condition which applies in particular to $X=BG$ for $G$ a compact Lie group) then for each $n$ the map $H^n(X)\to H^n(X;\mathbb{R})$ is injective if and only if $H_{n-1}(X)$ is torsion free. Here and below integer coefficients are omitted from the notation.

To see this, note that the universal coefficient sequence $$ 0\to \operatorname{Ext}(H_{n-1}(X),A)\to H^n(X;A)\to \operatorname{Hom}(H_n(X),A)\to 0 $$ is natural with respect to homomorphisms $A\to A'$ of abelian coefficient groups. If $H_{n-1}(X)$ is torsion free, the ext groups $\operatorname{Ext}(H_{n-1}(X),\mathbb{Z})$ and $\operatorname{Ext}(H_{n-1}(X),\mathbb{R})$ both vanish, and the map $H^n(X)\to H^n(X;\mathbb{R})$ is injective if and only if $\operatorname{Hom}(H_n(X),\mathbb{Z})\to \operatorname{Hom}(H_n(X),\mathbb{R})$ is injective (which it clearly is).

Conversely, if $H_{n-1}(X)$ has torsion then so does $H^n(X)$, and this torsion is in the kernel of $H^n(X)\to H^n(X;\mathbb{R})$.

This is also equivalent to the corresponding statement for homology: Since $Hom(C,\mathbb{Z}) \otimes \mathbb{R} = Hom(C,\mathbb{R})$ for $C$ a free abelian group, the cochain complex with $\mathbb{R}$-coefficients is isomorphic to the cochain complex with $\mathbb{Z}$-coefficients tensored with $\mathbb{R}$. The same homological algebra as in the homology universal coefficient theorem gives you that $H^n(X) \rightarrow H^n(X; \mathbb{R})$ precisely kills torsion. (Which of course comes from $H_{n-1}$ if you invoke the universal coefficient sequence as in Marks answer.)