Assuming that there is an $\omega_1$-dense $\sigma$-complete ideal, $I$, on $\omega_1$ (such that every positive set can be partitioned into $\omega_1$ positive sets), we can generalize Joel's construction to $\lambda=\omega_1$ and prove that this forcing collapses $2^{\aleph_1}$.
Let $p = \langle p_\alpha \mid \alpha < \omega_1\rangle$ be a condition. Let $A$ be the dense set in $I^+$ of size $\omega_1$. Let $r\in V$ be a real that codes an index of some $C\in A$, and $n<\omega$ such that $B=\{\alpha\in C\mid n\in\text{dom }p_\alpha\}\in I$. There is such $C, n$ since $I$ is $\sigma$-complete.
Split $C\setminus B$ into $\omega_1$ sets from $A$, up to an error in $I$, $\{F_i \mid i < \omega_1\}$. We can assume that for every $i < j < \omega_1$, $F_i \cap F_j = \emptyset$ (using the $\sigma$-completeness). Code an element of $y \in \mathcal P(\omega_1)$ by coloring the components: $q\leq p$, $q_\alpha (n) = 1$ iff $\alpha \in F_i$ where $i \in y$. Now, every $y \in \mathcal{P}(\omega_1)$ is coded by a real from $V$ which in turn coded by a natural number.
Assuming that there is also an $\omega_2$-complete $\omega_2$-dense ideal on $\omega_2$ we can continue to $\lambda = \omega_2$: code a real, use it to code subset of $\omega_1$ which translate to an element in the dense subset corresponding to the ideal on $\omega_2$. Split it and code by the same method a subset of $\omega_2$. Assuming that those dense ideals exist for every regular $\kappa$ (which is consistent relative to a huge cardinal, by a theorem of Foreman), it is possible to continue with this method up to the first inaccessible. Foreman's theorem only gives us a $\kappa$-centered ideal on $\kappa$, but similar argument works for those ideals as well.
It is not clear at all if we actually need the large cardinals here.