I believe I can show that a partition lattice of a separably-acting type II or III factor is forcing-equivalent to the usual (Levy) continuum-collapsing poset, on the supposition that each of its nontrivial partitions has continuum cardinality. I suspect this is always the case, and maybe it can be proved in an elegant way, but for now I can only prove that a type III projection lattice has no countable nontrivial partition, so I need the continuum hypothesis. My no-countable-nontrivial-partition proof is not very elegant or edifying so I won't post it, but I mention the main idea in a comment here: When are there countable partitions in a factor's projection lattice?
$\mathcal{R}$ is a separably-acting type II or III von Neumann factor; $\mathbb{P}$ is its projection lattice,
$ \mathbb{P} = \{ P \in \mathcal{R} : P^2 = P^* = P \} , $
with the usual ordering $P \leq Q \iff PQ = P$. The greatest and least members of $\mathbb{P}$ are the identity projection $1$ and the null projection $0$. $\mathbb{P}^+$ means $\mathbb{P} \setminus \{ 0 \}$. When $A, B \in \mathbb{P}$, we write $A \perp B$ to mean they are orthogonal ($AB = 0$).
By a partition of $P \in \mathbb{P}$ we mean a lattice partition, i.e. a maximal pairwise-disjoint subset of the projections in $\mathbb{P}^+$ that are $\leq P$. A partition is nontrivial if its cardinality is $> 1$.
A state $\phi$ on $\mathcal{R}$ is a complex linear functional on $\mathcal{R}$ that is normalized in the sense that $\phi(1) = 1$. A state is faithful if $\phi(T) = 0$ holds only when $T = 0$; it is normal if it is countably additive on sets of mutually orthogonal projections, in which case it is also continuous on $\mathbb{P}$ with respect to the strong operator topology.
For cardinals $\kappa > \aleph_0$, $C_{\kappa}$ is the set of finite sequences of ordinals $< \kappa$, ordered by reverse inclusion; this is the standard poset used to force $\kappa$ to become countable.
Recall that when $A, B$ are posets, a dense embedding $\phi : A \rightarrow B$ is an order-isomorphism onto a subset of $B$ such that every $b \in B$ has some $a \in A$ satisfying $\phi(a) \leq b$.
If one separative poset embeds densely into another, then the embedding extends naturally to an isomorphism between their boolean completions. Thus to show $B(C_{\kappa})$ isomorphic to $B(\mathbb{P})$, it suffices to show that $C_{\kappa}$ embeds densely into $\mathbb{P}^+$.
Lemma 1: Suppose that for some faithful normal state $\phi$ on $\mathcal{R}$, and all $P \in \mathbb{P}^+$, and all $\epsilon > 0$, there exists a lattice partition $X$ of $P$ such that (i) $|X| = 2^{\aleph_0}$, (ii) $Q \in X \Rightarrow \phi(Q) < \epsilon$, and (iii) for all $P' \in \mathbb{P}^+$, $P' \leq P$, there exists $Q \in X$ such that either $P' \leq Q$ or $Q \leq P'$. Then there is a dense embedding of $C_{2^{\aleph_0}}$ into $\mathbb{P}^+$.
Proof: Let $P_{ \langle \rangle }$ (indexed by the empty sequence) be $1 \in \mathbb{P}$. Let $\{ P_{\langle \alpha \rangle} : \alpha < 2^{\omega_0} \}$ be a lattice partition of $P_{\langle \rangle}$ of the kind supposed, with $\epsilon = 1/2$. Next, for each $\alpha$ obtain a lattice partition $\{ P_{\langle \alpha, \beta \rangle} : \beta < 2^{\omega_0} \}$ of $P_{\langle \alpha \rangle}$, with $\epsilon = 1/4$. Continue in this manner so that for every finite sequence $\vec{\alpha}$ of ordinals $< 2^{\omega_0}$, $\phi( P_{\vec{\alpha}} ) < 2^{ - |\vec{\alpha}|}$. Clearly $\vec{\alpha} \mapsto P_{\vec{\alpha}}$ is an embedding of $C_{2^{\aleph_0}}$ into $\mathbb{P}^+$; we must confirm that it is a dense embedding.
For $n > 0$, define $X_n := \{ P_{\vec{\alpha}} : | \vec{\alpha} | = n \}$. Each $X_n$ is then a lattice partition of $1 \in \mathbb{P}$ satisfying (i), (ii), (iii) for $\epsilon = 2^{ - n}$. Now given $P \in \mathbb{P}^+$, fix $n$ such that $2^{-n} < \phi(P)$; then by (iii), some $P_{\vec{\alpha}} \in X_n$ satisfies either $P \leq P_{\vec{\alpha}}$ or $P_{\vec{\alpha}} \leq P$; but the first alternative is excluded because each $P_{\vec{\alpha}} \in X_n$ satisfies $\phi(P_{\vec{\alpha}}) < 2^{-n} < \phi(P)$.
Lemma 2: If, for all $P \in \mathbb{P}^+$, every nontrivial lattice partition of $P$ has cardinality $2^{\aleph_0}$, then there is a dense embedding of $C_{2^{\aleph_0}}$ into $\mathbb{P}^+$.
Proof: Fix $P \in \mathbb{P}^+$, $\epsilon > 0$, and a faithful normal state $\phi$ on $\mathcal{R}$; it suffices to show there exists a lattice partition $X$ of $P$ meeting requirements (i), (ii), (iii) of Lemma 1.
Enumerate as $\{ P_\alpha : \alpha < 2^{\aleph_0} \}$ the projections in $\mathbb{P}^+$ that are $< P$.
We define $Q_\alpha$ by induction on $\alpha$, for all $\alpha < 2^{\aleph_0}$, with the induction hypothesis that the previously-defined $Q_\beta$ are pairwise-disjoint. At step $\alpha$, if $P_\alpha \leq Q_\beta$ for some $\beta < \alpha$, let $Q_\alpha = 0$. Otherwise, under the present lemma's supposition, $\{ Q_\beta \wedge P_\alpha : \beta < \alpha \}$ cannot be a partition of $P_\alpha$ (in the lattice $\mathbb{P}$), because its cardinality is less than $2^{\aleph_0}$. Thus $\{ Q_\beta \wedge P_\alpha : \beta < \alpha \}$ is a pairwise-disjoint but not maximal pairwise-disjoint subset of the projections in $\mathbb{P}^+$ below $P_\alpha$; so there exists in $\mathbb{P}^+$ a projection $< P_\alpha$ that is disjoint from all $Q_\beta$, $\beta < \alpha$. Let $Q_\alpha$ be such a projection satisfying $\phi(Q_\alpha) < \epsilon$ (it follows the fact that these projections form a downward-closed subset of $\mathbb{P}^+$, which has no minimal projections, and from the normality of $\phi$, that we can require an arbitrarily small $\phi$ value here).
Let $X$ be the set of all the nonzero $Q_\alpha$ so obtained. Clearly requirements (ii) and (iii) of Lemma 1 are met, and $X$ is pairwise-disjoint. Moreover $X$ must be a maximal pairwise-disjoint subset of the projections in $\mathbb{P}^+$ below $P$ -- i.e. a partition of $P$ -- because any such projection we might hope to add to $X$ was already enumerated as one of the $P_\alpha$, and either $P_\alpha$ was $\leq Q_\beta$ for some $\beta < \alpha$ or $P_\alpha \wedge Q_\alpha \neq 0$. Finally, by supposition, $X$ has the cardinality required by (i).
Corollary: If CH holds and $\mathbb{P}$ has no countable nontrivial partition then there is a dense embedding of $C_{2^{\aleph_0}}$ into $\mathbb{P}^+$.
Proof: CH means that $2^{\aleph_0}$ is $\aleph_1$, the smallest uncountable ordinal. This plus the supposition that $\mathbb{P}$'s nontrivial partitions are uncountable implies they all have cardinality $2^{\aleph_0}$ (note $| \mathbb{P} | \leq | \mathcal{R} | \leq | \mathcal{B}(H) |$, which is $\leq 2^{\aleph_0}$ when $H$ is separable, because $\mathcal{B}(H)$ then has a countable dense subset in the strong operator topology, from which at most $2^{\aleph_0}$ convergent nets can be chosen). And this, plus the observation that $\{ P' \in \mathbb{P} : P' \leq P \}$ is order-isomorphic to all of $\mathbb{P}$ for any $P \in \mathbb{P}^+$, implies the supposition of Lemma 2.
