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Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?

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No. If $P$ is any set and $<$ is the empty relation then the interval topology is just the cofinite topology. Now if $P$ has size $\mathfrak c$ then $P$ with the cofinite topology is path-connected just because any biyection $[0,1] \to P$ is continuous.

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