I suppose you are talking about deRham cohomology. Then it would be wise to take a look at the work of Richard Melrose, e.g. his book on the APS theorem:
http://www-math.mit.edu/~rbm/book.html
On page 65 he discusses deRham cohomology for manifolds with boundary (which can be easily generalized to the corner case, as was also done by him!).
On manifolds with corners something interesting happens - there are different versions of reasonable vector fields (and - by duality - differential forms  ), e.g.

1. extendible vector fields (like you mentioned)
2. tangent vector fields (tangent to any boundary hypersurface)
3. "zero" vector fields (vanishing on all boundary hypersurfaces)
   (It can be shown that d preserves the classes 1.-3., giving a deRham complex whose cohomology can be computed) Melrose points out (compact with boundary case) that in cases 1. and 3. the deRham cohomology is canonically isomorphic to the singular cohomology of the underlying topological space. For the 2nd case also the cohomology of the boundary enters (via a degree shift).

I should also point out that there is also a working Morse theory on manifolds with corners, see for example
http://www.springerlink.com/content/m20983xw42015t35/
and
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1K-45TY6KH-1&_user=10&_coverDate=11%2F30%2F2002&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1188262351&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8b2c4ec9a50cdb034dbdd9f078f0335b

Furthermore it is easy to construct "invariants" of the manifold with corners by also taking into account its corners (but be careful w.r.t. which transformations this is an invariant)!

I suppose you are talking about deRham cohomology. Then it would be wise to take a look at the work of Richard Melrose, e.g. his book The Atiyah-Patodi-Singer Index Theorem.

On page 65 he discusses deRham cohomology for manifolds with boundary (which can be easily generalized to the corner case, as was also done by him!). On manifolds with corners something interesting happens: there are different versions of reasonable vector fields (and—by duality—differential forms), e.g.

1. extendible vector fields (like you mentioned)
2. tangent vector fields (tangent to any boundary hypersurface)
3. "zero" vector fields (vanishing on all boundary hypersurfaces)

(It can be shown that $d$ preserves the classes 1.-3., giving a deRham complex whose cohomology can be computed) Melrose points out (compact with boundary case) that in cases 1. and 3. the deRham cohomology is canonically isomorphic to the singular cohomology of the underlying topological space. For the 2nd case also the cohomology of the boundary enters (via a degree shift).

I should also point out that there is also a working Morse theory on manifolds with corners, see for example

M. Shida, Fundamental Theorems of Morse Theory for Optimization on Manifolds with Corners, Journal of Optimization Theory and Applications 106 (2000) pp 683-688, doi:10.1023/A:1004669815654

and [this broken link EDIT perhaps someone else can extract a result -DR]

Furthermore it is easy to construct "invariants" of the manifold with corners by also taking into account its corners (but be careful with respect to which transformations this is an invariant)!