Problem: Find an example of a positive function $f: [0,1] \to \mathbb{R}_{>0}$ that is of bounded variation, whose reciprocal $1/f$ is integrable but not of bounded variation.
One necessary condition for $f$ is that $\inf_{x \in [0,1]} f(x)=0$, but I don't know how to proceed further.
You have more or less resolved this with the observation that $\inf_{x \in [0,1]} f(x)=0$. Just take the simplest such function, e.g. $$f(x)= \begin{cases}\sqrt{(x)},\qquad x\neq0,\\1,\qquad x=0.\end{cases}$$
Apart from your observation, the only consideration is making sure that the reciprocal is integrable.
Note that any continuous $f$ with the property $\inf_{x \in [0,1]} f(x)=0$, will not be positive.