How to add animated gradient to an svg path?

You can use the SMIL animation of SVG. The idea is to animate the color-stop or the offset of the gradient to create the needed effect:

svg {
 border:1px solid;
 width:200px;
}

<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" 
   viewBox="10 10 24 24">
    <defs>
        <linearGradient id="linear-gradient" x1="-100%" y1="0" x2="200%" y2="0" >
            <stop offset="0" stop-color="red">
               <animate attributeName="offset" values="0;0.2;0.5" dur="2s" repeatCount="indefinite"  /> 
            </stop>
            <stop offset="0.5" stop-color="yellow">
                <animate attributeName="offset" values="0.5;0.7;0.8;1" dur="2s" repeatCount="indefinite"  /> 
            </stop>
        </linearGradient>
    </defs>
<path fill="url(#linear-gradient)" d="M20,31 C15.4189994,27.2225585 12.5023327,24.2225585 11.25,22 C10.2743515,20.6156479 10,19.6181623 10,18.1428571 C10,15.5113854 12.4883456,13 15,13 C17.3176009,13 18.9621484,13.8491346 20,15.5714286 C21.0382977,13.8491346 22.6828452,13 25,13 C27.5116544,13 30,15.5113854 30,18.1428571 C30,19.6181623 29.7256485,20.6156479 28.75,22 C27.497816,24.2225585 24.5811493,27.2225585 20,31 Z" />
</svg>

Animating the color:

svg {
 border:1px solid;
 width:200px;
}

<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" 
   viewBox="10 10 24 24">
    <defs>
        <linearGradient id="linear-gradient" x1="0" y1="0" x2="100%" y2="0" >
            <stop offset="0" stop-color="red">
               <animate attributeName="stop-color" values="yellow;red" dur="2s" repeatCount="indefinite"  /> 
            </stop>
            <stop offset="1" stop-color="yellow">
                <animate attributeName="stop-color" values="red;yellow" dur="2s" repeatCount="indefinite"  /> 
            </stop>
        </linearGradient>
    </defs>
<path fill="url(#linear-gradient)" d="M20,31 C15.4189994,27.2225585 12.5023327,24.2225585 11.25,22 C10.2743515,20.6156479 10,19.6181623 10,18.1428571 C10,15.5113854 12.4883456,13 15,13 C17.3176009,13 18.9621484,13.8491346 20,15.5714286 C21.0382977,13.8491346 22.6828452,13 25,13 C27.5116544,13 30,15.5113854 30,18.1428571 C30,19.6181623 29.7256485,20.6156479 28.75,22 C27.497816,24.2225585 24.5811493,27.2225585 20,31 Z" />
</svg>

Another idea is to consider the path inside a mask then you run a CSS animation to easily animate the background properties:

svg {
  border: 1px solid;
  width: 200px;
}

.box {
  width:200px;
  height:200px;
  background:linear-gradient(to right,red,green,blue) left/200% 100%;
  -webkit-mask:url('data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg"  viewBox="10 10 24 24"><path  d="M20,31 C15.4189994,27.2225585 12.5023327,24.2225585 11.25,22 C10.2743515,20.6156479 10,19.6181623 10,18.1428571 C10,15.5113854 12.4883456,13 15,13 C17.3176009,13 18.9621484,13.8491346 20,15.5714286 C21.0382977,13.8491346 22.6828452,13 25,13 C27.5116544,13 30,15.5113854 30,18.1428571 C30,19.6181623 29.7256485,20.6156479 28.75,22 C27.497816,24.2225585 24.5811493,27.2225585 20,31 Z" /></svg>') center/contain;
          mask:url('data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg"  viewBox="10 10 24 24"><path  d="M20,31 C15.4189994,27.2225585 12.5023327,24.2225585 11.25,22 C10.2743515,20.6156479 10,19.6181623 10,18.1428571 C10,15.5113854 12.4883456,13 15,13 C17.3176009,13 18.9621484,13.8491346 20,15.5714286 C21.0382977,13.8491346 22.6828452,13 25,13 C27.5116544,13 30,15.5113854 30,18.1428571 C30,19.6181623 29.7256485,20.6156479 28.75,22 C27.497816,24.2225585 24.5811493,27.2225585 20,31 Z" /></svg>') center/contain;
          
   animation:change 2s infinite linear alternate;
}
@keyframes change {
  to {
    background-position:right;
  }
}

<div class="box">
</div>

Related question to get more detail about background calculation: Using percentage values with background-position on a linear-gradient

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