3 added 400 characters in body

The simplest smooth curve that avoids all rational points is probably the parabola

$$y = \frac{b}{a}x + \lambda x(a-x)$$

where $\lambda$ is any irrational number.

Now, the set of points in $\mathbb{R}^2$ that are a rational distance from both $o$ and $p$ is countable (because any two rationals determine at most two such points); but the set of irrational $\lambda$ is uncountable. Moreover, as $\lambda$ varies, the resulting parabolas are all disjoint (apart from their end-points $o$ and $p$). So there must exist (uncountably many) $\lambda$ for which the above parabola satisfies your conditions. Also, as you suspect, $\lambda$ can be chosen as close to zero as you like.

Updated In fact we can take any one-parameter family of curves ${C_{\lambda}}$ from $o$ to $p$ which are disjoint except at $o$ and $p$. The set of all rational points, together with the set of points that are a rational distance from both $o$ and $p$, is countable, so in any open interval $I$ there exists $\lambda \in I$ such that ${C_{\lambda}}$ satisfies OP's conditions (a) and (b).

The simplest smooth curve that avoids all rational points is probably the parabola

$$y = \frac{b}{a}x + \lambda x(a-x)$$

where $\lambda$ is any irrational number.

As Tsuyoshi Ito

Now, the set of points out in $\mathbb{R}^2$ that are a comment, it rational distance from both $o$ and $p$ is impossible countable (because any two rationals determine at most two such points); but the set of irrational $\lambda$ is uncountable. Moreover, as $\lambda$ varies, the resulting parabolas are all disjoint (apart from their end-points $o$ and $p$). So there must exist (uncountably many) $\lambda$ for which the above parabola satisfies your conditions. Also, as you suspect, $\lambda$ can be chosen as close to avoid rational distanceszero as you like.

1

The simplest smooth curve that avoids all rational points is probably the parabola

$$y = \frac{b}{a}x + \lambda x(a-x)$$

where $\lambda$ is any irrational number.

As Tsuyoshi Ito points out in a comment, it is impossible to avoid rational distances.