$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ .
Clearly the height of primes in support of $H^i_I(R)$ is at least $i$
The question is if it contains a prime of height $i$, specially when $R$ is complete and unmixed?
PS What about the case $i$ equals the cohomological dimension of $I$, ie
$$i=\sup\lbrace j: H^j_I(R)\neq 0\rbrace $$