Is the Lorentz group $O(1,3)$ irreducible, i.e. simple?

Groups are not reducible or irreducible, representations are.

You are thinking of the notion of simplicity vs. semi-simplicity.

Let me discuss it at the level of Lie algebras so we don't have any global issues: the Lie algebra $\mathfrak{so}(1,3)$ is semisimple but it is not simple because, as you noted, it is the direct sum of two algebras $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$.

A consequence of this is that the adjoint representation is reducible, indeed it is $\mathbf{adj} = (1,0)\oplus(0,1)$. In general $$ \mbox{$\mathfrak{g}$ is simple} \;\Longleftrightarrow\; \mbox{$\mathbf{adj}(\mathfrak{g})$ is irreducible} $$

Nevertheless, non-simple groups can and do have irreducible representations.

For what it's worth:

• The real Lorentz Lie algebra $so(1,3;\mathbb{R})\cong sl(2,\mathbb{C})$ is simple.

• Its complexification $so(1,3;\mathbb{C})\cong sl(2,\mathbb{C}\oplus sl(2,\mathbb{C})$ is semisimple but not simple.

See also this related Phys.SE post.