This is comment rather than answer:  Please check it, whether it makes sense...

Corollary 2.6 page 11 of [Free Loop space and homology by J.L Loday][1] says that 
For any simply  connected space, there is a functorial isomorphism:
$$HH_1 (\Omega^1(M)) \cong H^1(LM)$$
And  [Hochschild-Kostant-Rosenberg theorem][2]  says that: For a k-algebra $R$, its module of Kähler differentials coincides with its first Hochschild homology
$$\Omega_1(R/k)\cong HH_1(R)$$

Now we have by this [MO post][3], a surjective map $\Omega_1(C^\infty(M))\to \Omega^1(M)$. 

So can we say that $H^1(LM)= \Omega_1(C^\infty(M))$ and if $\Omega^1(M)\neq \{0\}$, we have $H^1(LM)\neq \{0\}$ for simply connected finite dimension manifold $M$.


  [1]: http://www-irma.u-strasbg.fr/~loday/PAPERS/FreeLoop4.pdf
  [2]: http://ncatlab.org/nlab/show/Hochschild-Kostant-Rosenberg+theorem
  [3]: http://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials/6138#6138