It depends on how you define orthogonality, or, as OSE puts it in his comment, "Orthogonality is usually tested using some defined inner product." I'll expand on this a bit.
In order to mathematically answer the question
Is direction A orthogonal to direction B?
we need a definition of the terms "direction" and "orthogonal." The standard mathematical way to formalize the notion of direction is by using vectors.
For example, imagine we are traveling along some curve $\gamma$ in the plane, and let's say we are at some point $x$ in the plane, then the direction of $\gamma$ at the point $x$ can be defined by a vector tangent to $\gamma$ at the point $x$.
In particular, let's say that $\gamma$ is just the $x$-axis, then a tangent vector to the $x$-axis at every point is just $(1,0)$ (or any positive scalar multiple of this), and this defines the direction of this line at every point. Similarly, the direction of the $y$-axis at every point is defined by the vector $(0,1)$ (or any positive scalar multiple of this).
What about the notion of orthogonality? Well since vectors define directions, we might be inclined to think that orthogonality of directions should be defined in terms of associating a number to each pair of vectors, and that when this number has a special value, we call these vectors (and therefore the directions they define) orthogonal.
In practice, that's exactly how it's done. The association of a number to a given pair of vectors that tests for orthogonality is called an inner product, as mentioned in OSE's comment to your question. Given any pair of vectors $u$ and $v$, it is common to see the inner product denoted by something like $u\cdot v$, or $\langle u,v\rangle$, or something similar, depending on the context. Given an inner product, two vectors are said to be orthogonal with respect to that inner product provided their inner product is zero.
So let's take the example of directions in the plane. The standard inner product on the plane, often referred to as the "dot product" is defined as follows:
\begin{align}
(u_x, u_y)\cdot (v_x, v_y) = u_xu_y + v_xv_y
\end{align}
To test that two directions are orthogonal, we just need to take their inner product and verify that it's zero. For example, the $x$- and $y$- directions are orthogonal since
\begin{align}
(1,0)\cdot (0,1) = 1(0) + 0(1) = 0.
\end{align}
Now let's go back to the original question of time and space being orthogonal. Let's restrict the discussion to $1+1$-dimensional spacetime with coordinates $(t,x)$ for simplicity. The direction of the time axis is given by the unit vector $(1,0)$. The direction of the space axis is given by the unit vector $(0,1)$. Calling them orthogonal now depends on the inner product we specify.
If we choose the inner product to be just like the dot product on the $x$-$y$ plane, namely if we choose
\begin{align}
(u_t, u_x)\cdot (v_t, v_x) = u_tv_t + u_xv_x,
\end{align}
then yes, time and space are orthogonal with respect to this product.
However, the physical interpretation and significance of applying this inner product to spacetime is murky. The standard inner product on the plane is motivated by the fact that it comports with the usual notion of distance. In particular, if two vectors are orthogonal with respect to this inner product, then the sum of the square of their lengths agrees with the independently defined notion of the Euclidean distance between their endpoints.
In the case of spacetime, this notion of distance isn't particularly useful or appropriate. There is, however a different notion of "distance" derived from a scalar product (which is not strictly speaking an inner product since it's not positive definite) defined by
\begin{align}
(u_t, u_x)\cdot (v_t, v_x) = -u_tv_t + u_xv_x.
\end{align}
Unfortunately, since this scalar product is not an inner product, the notion of orthogonality is rather strained. But if you insist on still calling vectors orthogonal if their scalar product with each other is zero, then the $x$ and $t$ directions are still orthogonal relative to this product.
