Hal Schenck kindly provided a counter-example to the claim in question. Consider the twisted cubic: (the code is in Macaulay)
o3 I = ideal (- y^2 + xz, - yz + xw, - z^2 + yw)
i4 : J = ideal leadTerm I
o4 = ideal (z^2 , y*z, y^2 )
i5 : hilbertPolynomial coker gens ideal leadTerm I^2
o5 = - 16P + 9P 0 1
i6 : hilbertPolynomial coker gens ideal leadTerm J^2
o6 = - 20P_0 + 10P_1
So, the quotient is actually positive dimensional.