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 assume that the collection $\{F_i \mid i < \omega_1\}$ is the first such partition in some fixed well ordering. 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. 

The decoding is as follows: Let $G$ be the generic filter. Let $g_\alpha \colon \omega \to \omega$ be the generic real defined by $G$ for $\alpha < \omega_1$. For every natural $m$ we check if $n$ codes a real number in the means of Joel's answer. Assume that it does, and let $r_m$ be this real. By fixing a surjection from $2^\omega$ onto $A \times \omega$, $r$ corresponds to a couple $(C_m, n_m)$. Let $\{ F^m_i \mid i < \omega_1\}$ be the partition as above. Let:
 $$y_m = \{i < \omega_1 \mid \{\alpha \in F_i \mid g_\alpha (n) = 0\} = F_i \mod I\}$$.
By density arguments, we know that for every $y\in \mathcal{P}(\omega_1)^V$ there is $m$ such that $y = y_m$. 

**Generalization:** In order to generalize to higher cardinals (up to the first inaccessible), we will need to use $\kappa$-centered ideals on $\kappa$ (instead of $\kappa$-dense). An ideal $I$ is $\kappa$-dense if there are collection of $\kappa$ filters $\mathcal{F}_\alpha$, $\alpha < \kappa$ such that $I^+ = \bigcup_\alpha \mathcal{F}_\alpha$.  

By a theorem of Foreman (appears in the Handbook, page 1047, Theorem 7.57), it is consistent relative to a huge cardinal that every regular cardinal $\kappa$ carries a $\kappa$-complete $\kappa$-centered ideal. Moreover, for accessible $\kappa$, those ideals (and also their restriction) are not $\kappa$-saturated, so every positive set $A$ can be partitioned into $\kappa$ positive sets. 

Let's assume that in $V$ for every regular $\kappa$, there is $\kappa$-centered $\kappa$-complete non-trivial ideal. Assume also $GCH$, for simplicity.

We will show by induction on cardinals $\lambda$ less than the first inaccessible (if there is one), that the product of $\lambda$ copies of the Cohen forcing collapses $2^{\lambda}$. In fact we will construct the coding method (relatively) explicitly. 

**Case 1:** $\lambda = \mu^+$. This is similar to the $\omega_1$ case above. Let $I$ be a $\lambda$-centered, $\lambda$-complete ideal on $\lambda$ as witnessed by $\{\mathcal{F}_i \mid i < \lambda\}$. We will generically code every element $y$ of $2^\lambda$ by an ordinal from $\lambda$ and a natural number (which in turn are generically coded by a natural number, by the induction hypothesis). Let $p$ be a condition and assume that $n$ is a natural number such that $C = \{\alpha < \lambda \mid n\notin \text{dom }p_\alpha\}\in I^+$. Assume that this set is in $\mathcal{F}_\gamma$. Since the difference between all two sets in $\mathcal{F}_\gamma$ is in $I$, we can pick an arbitrary element there and split it, as before, to $\lambda$ sets - $\{F_i \mid i < \lambda\}$. Again, those sets can be assumed to be disjoint and their union is $C$, up to an element from $I$. We continue, and extend $p$ to code $y$ in the generic in the same way as above: $q\leq p$, $q_\alpha(n) = 0$ iff $\alpha \in F_i \cap C$ and $i\in y$ so $y$ will be coded by $(\gamma, n)$ in the generic filter.

**Case 2:** $\lambda$ is singular. Let $\{\mu_i\mid i < \text{cf }\lambda\}$ be a cofinal sequence, $\mu_0 > \text{cf }\lambda$. We split the product into a product of $\text{cf } \lambda$ parts. It is clear that every element of $2^\lambda = \prod 2^{\mu_i}$ is coded by an element in $(\omega^{\text{cf }\lambda})^V$, but the latter is coded with a natural number, by the induction hypothesis. 

It is not clear at all if we actually need the large cardinals here.