Is inverse matched filter useful? I am aware of matched filter and its application. Now, wondering if there is any application of inverse matched filter? What I mean my inverse matched filter is that convolution of matched filter and the inverse matched filter would lead to close to delta function.
In your question, you postulate the existence of two filters, let's say $p(t)$ and $g(t)$, such that $p(t) \star g(t)=\delta(t)$. This assumption is problematic because it implies that $G(f)=1/P(f)$, which requires $P(f) \neq 0$ for all $f$. Furthermore, for small values of $P(f)$, $G(f)$ will take arbitrarily large values (see Dilip Sarwate's comments above and below this answer).

Assuming (for the sake of discussion) the existance of such a pair of filters, there may not be any advantage to using them. I'll describe a possible application from digital communications. Let's say you want to transmit a number $A$ over an analog channel. You choose an appropriate analog pulse $p(t)$ and transmit $$s(t) = A\delta(t)\star p(t).$$ (The pulse $p(t)$ could be chosen to fit the channel bandwidth, and/or to have a certain energy). The receiver could perform the "inverse filter" operation on the signal $s(t)$: $$s(t) \star g(t)=A\delta(t),$$ where $g(t)$ is your "inverse matched filter". However, in practice it turns out that detecting $A$ in this way is not optimal, because it does not have the best signal-to-noise ratio. In other words, when you add noise to the transmitted signal, the received signal becomes $$r(t)=s(t)+n(t).$$ In this case, filtering $r(t)$ with the "inverse matched filter" $g(t)$ results in a worse signal-to-noise ratio than filtering with the filter matched to $p(t)$.

If you have access to it, I highly recommend "An Introduction to Matched Filters", by G. Turin, IRE Transactions on Information Theory, June 1960. Your question is answered in page 318.