Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and attaching each $X_i\times \{1\}$ to $X_{i+1}\times \{0\}$ via the map $X_i\to X_{i+1}$.
What is an example (if there is one) in which the homotopy direct limit of such a sequence is not homotopy equivalent to the regular direct limit (in the category of topological spaces) of that sequence?
If your maps are cofibrations, then the direct limit and the homotopy direct limit coincide (up to homotopy equivalence). So to find a counterexample, we must look at some less nice sequences of maps.
For instance, consider the map $f : S^1 \to S^1$ defined by $e^{\theta i} \mapsto e^{2 \theta i}$. This is not a cofibration, obviously. The direct limit of the sequence $$S^1 \stackrel{f}{\to} S^1 \stackrel{f}{\to} S^1 \stackrel{f}{\to} S^1 \to \cdots$$ is the quotient of $S^1$ as a topological group by the subgroup $G = \{ e^{2^{-n} k \pi i} : n \in \mathbb{N}, k \in \mathbb{N} \}$. This rather monstrous-seeming space is actually the indiscrete topological space on $\left| S^1 \right|$-many points, hence is contractible.
On the other hand, the homotopy direct limit of the sequence is not contractible. Indeed, $\pi_1$ sends homotopy direct limits to direct limits, and the corresponding sequence of $\pi_1$ is $$\mathbb{Z} \stackrel{2 \times}{\to} \mathbb{Z} \stackrel{2 \times}{\to} \mathbb{Z} \stackrel{2 \times}{\to} \mathbb{Z} \to \cdots$$ so $\pi_1$ of the direct limit is additive group of the ring $\mathbb{Z} [\frac{1}{2}]$.