Many points with rational coordinates are known with rational distances to three vertices of a unit square. For example, the following points are rational distances from $a=(0,0)$, $b=(1,0)$, and $c=(0,1)$.
$(945/3364, 225/841)$, $(99/175, 297/700)$, $(8288/12675, 1628/4225)$, $(1155/10952, 99/2738)$
Is there a point with irrational coordinates that has rational distances to points $a, b, c$?
On
I would say no consider only two points which form an ellipse then by the erdos anning theorem all points of mutual rational distance (ellipse) which includes the other point. all points then have rational coordinates.
this is poorly worded take the first two points as foci. take the radius such that the ellipse hits the third point.
From your comment below, you apparently take a point with irrational coordinates to mean a point with at least one irrational coordinate.
Suppose the point $(x,y)$ is a rational distance from all of $a,b,c$. Then the square of its distance from $a$ is $x^2+y^2$ and the square of its distance from $b$ is $(x-1)^2+y^2$. These must both be rational and hence also their difference $2x-1$. So $x$ is rational.
Similarly, the square of its distance from $a$ and the square of its distance from $c$ are both rational and hence also their difference $2y-1$. So $y$ is rational. Hence the point has rational coordinates.