Conceptually confused by LPC for speech: Do we synthesize by the inverse filter (FIR)?
Given speech samples, the LPC computation yields linear prediction coefficients $a_1,\ldots,a_p$. These describe a dependence model for a few samples. It is assumed that for short-enough collections of speech samples, the process is modeled well by an autoregressive stochastic process and that each sample is well-approximated by the following estimate based on the previous $p$ samples:
\begin{equation}
\widehat{x}[n] = a_1x[n-1] + \cdots + a_px[n-p].
\end{equation}
The error is then
\begin{equation}
e[n] = x[n] - a_1x[n-1] - \cdots - a_px[n-p].
\end{equation}
This "error filter" has coefficients $1, -a_1,\ldots,-a_p$.

For un-voiced speech, it turns out that discrete-time white noise is a decent model for the error signal. Keep this in mind.

Rather than send all the samples, communication systems send these linear prediction coefficients$^{\dagger}$. As mentioned in the other answer, to generate an approximation of the original speech, the synthesizer either generates a pulse train or a white noise realization and sends it though the filter that is the inverse of the error filter. This inverse is an IIR filter with frequency response
\begin{equation}
H(e^{i\omega}) = \frac{1}{1 - a_1e^{-i\omega} - \cdots - a_pe^{-ip\omega}}.
\end{equation}
The output of that IIR filter is an approximation of the original collection of speech samples.



$\dagger$ Vocoders do not really spit out linear prediction coefficients. They do not even spit out proxies such as reflection coefficients or line spectral pairs. They spit out integer indices of quantized versions of these coefficients. When these integers are received, the synthesizer grabs the appropriate entries in a lookup table and uses those as the coefficients.