Dependence and second Borel-Cantelli lemma.

I think Fristedt & Gray's book on probability proves a version of Borel--Cantelli that assumes only nonpositive correlation rather than independence. In particular, that means pairwise independence is enough.

Later edit: Here's what I find in A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray, page 79:

Lemma 5. [Borel-Cantelli] Let $(A_1,A_2,\ldots)$ be a sequence of events in a probability space $(\Omega,\mathcal{F},P)$. Assume that for each $i\ne j$, the events $A_i$ and $A_j$ are either negatively correlated or uncorrelated. Let $A=\lim\sup_{n\to\infty} A_n$. If $\sum_{n=1}^\infty P(A_n)=\infty,$ then $P(A)=1$.

The result you mention as being intuitive does not hold. To see this, one can recall the classical example of three uncorrelated dependent events, which is $A_1=[U=1]$, $A_2=[V=1]$ and $A_3=[UV=1]$, where $U$ and $V$ are i.i.d. $\pm1$ symmetric Bernoulli random variables.

Then the events $A_i$ are pairwise independent, each $A_i$ has probability $\frac12$, and $A_1\cap A_2\cap A_3=[U=V=1]$ hence $\mathrm P(A_1\cap A_2\cap A_3)=\frac14\gt\frac18=\mathrm P(A_1)\mathrm P(A_2)\mathrm P(A_3)$.