After enough searches, I've found the answer in the literature, at least for $y$ which is not too small w.r.t $x$. I would still appreciate input on smaller $y$, or on older references.

As is often the case in this kind of problems, this was answered by Tenenbaum and by Tenenbaum-Wu, who improve on earlier works by Song and Greaves.

The general problem that has been studied is the behaviour of $\sum_{n \le x, n \text{ is }y\text{-friable}} f(n)/n$ when $x \ge y \ge 2$, for multiplicative functions $f$ with $\sum_{p \le t} f(p) \log p$ close to $\kappa \log t$ for some $\kappa >0$. Given precise enough results, one can recover asymptotics for $\sum_{n > x, n \text{ is }y\text{-friable}} f(n)/n$, noticing that

$$\sum_{n \le x, n \text{ is }y\text{-friable}} f(n)/n + \sum_{n > x, n \text{ is }y\text{-friable}} f(n)/n = V_{f}(y), \qquad V_f(y) := \prod_{p \le y} \sum_{k \ge 0} f(p^k)/p^k.$$

I am only interested in $f \equiv \mathbf{1}$ (so $\kappa = 1$). If $x<y$, evidently
$$F(x,y) = V_{\mathbf{1}}(y) - \sum_{n \le x} 1/n \sim e^{\gamma} \log y \left( 1 - \frac{\log x}{e^{\gamma}\log y}\right)$$
by Mertens' third theorem. If $x \ge y$, we appeal to the main Theorem in Tenenbaum, ''Note on a paper by J. M. Song'', Acta Arith. 97 (2001), no. 4, 353–360, and find
$$F(x,y) = V_{\mathbf{1}}(y) \left(1- j_1( \log x/ \log y)(1+O(\log^{-\delta} y)) \right)$$
for every $\delta \in (0,1)$, where $j_1(u)$ is defined in terms of a delayed differential equation. In p. 4 of Tenenbaum and Wu, ''Moyennes de certaines fonctions multiplicatives sur les entiers friables. III'', Compos. Math. 144 (2008), no. 2, 339–376, our sum is denoted by $\psi^{*}_{\mathbf{1}}(x,y)$ and it is explained that
$$1- j_1(u) = \lambda_1(u), \qquad \lambda_1(u) := e^{-\gamma}\int_{u}^{\infty} \rho(v) \, d v$$
where $\rho$ is the Dickman function, and one has the large-$u$ asymptotics
$$\lambda_1(u) \sim \frac{e^{-\gamma}\rho(u)}{\log u}.$$
The error term $O(\log^{-\delta}y)$ in Tenenbaum can overtake the main term if $y$ is small w.r.t $x$; Theorem 3.2 in the aforementioned Tenenbaum-Wu paper improves the range.

To summarize,
$$ \frac{F(x,y)}{\log y} \asymp \frac{\rho(u)}{\log (u+1)}$$
as $x \to \infty$ if $y\ge 2$ is not too small w.r.t to $x$, where $u = \max\{ \log x/ \log y, 1\}$. For $u \to \infty$ this is an asymptotic result, while for bounded $u$ the constant is different.

This still leaves open the behaviour for small $y$. E.g. for $y=2$, $F(x,y)/\log y \asymp 1/x$ while $\rho(u)/\log(u+1)$ is much smaller. A naive guess is that the order of magnitude in this case is $F(x,y)/\log y \asymp \Psi(x,y)/(x\log x)$.