The answer has partly been given in the comments already, but I'll add a few things for clarity and for the sake of completeness.
First of all, when DSP engineers talk about IIR filters, they usually mean implementable (i.e., stable and causal) filters, which - apart from quantization effects - can be implemented exactly. In that sense, an ideal lowpass filter is not an IIR filter (because there exists no finite filter structure for implementing it), even though it clearly has an infinitely long impulse response. An ideal lowpass filter is a concept that we sometimes try to approximate by actually implementable filters. Note that an ideal lowpass is neither causal nor stable.
Second, when DSP engineers talk about lowpass filters, they usually mean filters that pass low frequencies with negligible distortion, and that attenuate higher frequencies sufficiently well. Any filter with that property is a lowpass filter. That's why there are FIR lowpass filters as well as IIR lowpass filters. Both FIR and IIR filters can approximate the ideal lowpass well enough for a given application. Note that we don't need to suppress the stopband components of the input signal more than the noise level. Also, magnitude and phase distortions in the passbands can always be made small enough such that they are tolerable for a given application.
In sum, there are FIR as well as IIR lowpass filters, and none of them equal the ideal (non-causal and unstable) lowpass filter. Which of the two is better completely depends on the application and on the platform. Everything we said about lowpass filters is true for all other filter types as well (highpass, bandpass, Hilbert transformer, etc.).