Let $(R,\mathfrak{m})$ be a Noetherian local ring, $M$ a non-zero finitely generated $R$-module, and let $I$ be an ideal of $R$. In this paper, we establish some new properties of local cohomology modules $\mathrm{H}^{i}_{I}(M)$, $i\geq 0$. In particular, we show that if $R$ is catenary, $M$ an equidimensional $R$-module of dimension $d$, and $x_{1},x_{2},\dots ,x_{t}$ is an $I$-filter regular sequence on $M$, then $(0:_{\mathrm{H}^{d-j}_{I}(\frac{M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})} x_{i})$ is $I$-cofinite for all $i = 1,2,\dots ,t$ and all $i \leq j \leq t$ if and only if $\mathrm{H}^{d-j}_{I}(\frac{M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})$ is $I$-cofinite for all $i = 1,2,\dots ,t$ and all $i \leq j \leq t$. Also we study the cofiniteness of local cohomology modules over homomorphic image of Cohen-Macaulay rings and we show that $\frac {\mathrm{H}^{\mathcal{W}(I,M)}_{I}(M)}{I\mathrm{H}^{\mathcal{W}(I,M)}_{I}(M)}$ has finite support, where $$\mathcal{W}(I,M) := \text{Max} \{i : \mathrm{H}^{i}_{I}(M) \text{~is not weakly Laskerian}\}.$$