Why can't charged objects exert electrostatic forces on heavier objects? We know that, for example, amber rubbed with fur attracts bits and pieces of straw, paper, Styrofoam, etc. which are all light objects. Why don't they attract heavier objects, likewise?
 A: Charged objects do attract uncharged heavy objects. If the weight of the object that is to be lifted is smaller than the force of electrostatic attraction, you will observe that the object will lift.
For instance,
A charged comb attracts our hair, but it doesn't mean that it will lift us even though our body's a conductor. We are just too heavy for that to happen. 
However , this effect is observable for our light hair.
Insight:
A charged body will always attract a uncharged body.
A: The charged objects attract uncharged objects by separating the charges on the uncharged object, that is the the charges of opposite nature to that of the charges on the charged object come closer. That is, the charged object induces opposite charges on the nearer surface of the uncharged object. We observe that light objects such as paper and straw lift up due to the electrostatic force of attraction. The similar attraction exists even for heavier objects. But they do not lift up because the electrostatic force is much less than their weight.
So, in summary, objects lift up only if their weight is smaller than than the electrostatic force of attraction.
A: They do attract heavier objects.
However the electrostatic force produced by a piece of rubbed amber is quite small because the net charge is quite small. To be able to pick up an object of mass $m$ the electrostatic force $F$ has to be greater than the gravitational force $mg$:
$$ F \gt mg $$
If the mass of the object, $m$, is large then the object is simply too heavy to be picked up. That's why only light objects like scraos of paper can be picked up.
A: The other answers are right: heavier objects are attracted, just due to their weight & inertia this isn't as noticable as with light objects.
However, this doesn't really answer the question IMO – it's only an answer if we presume that the electrostatic forces are the same for light an heavy objects, or, at least, that they scale less-than-proportionally with mass. But ab initio, it should seem just as plausible that they scale proportionally with mass (as would be the case if every atom brought some fixed net charge, in addition to its fixed net mass).
But in fact, the objects we're talking about are net neutral. They are, as TAJAS P says, attracted because an inhomogeneous field turns these neutrals into induced dipoles. And the strength of these dipoles is proportional to the length $\ell$ of the neutral object (measured in parallel with the vector pointing towards the charge), more or less regardless of the material properties†. More precisely, it is proportional to the difference in electric field strength between opposite ends of the object (which in first Taylor-expansion scales proportionally with object length). The force of attraction is also proportional to this field-strength difference, so we have
$$
  F \propto \ell\cdot\ell = \ell^2
$$
But force alone doesn't tell us much about what movements we'll see happening: it's acceleration $F/m$ that's important, and the mass of a voluminous object scales as
$$
  m = V \cdot \rho \propto \rho \cdot \ell^3
$$
where $\rho$ is the mass density. So all in all,
$$
  a \propto \frac{\ell^2}{\ell^3 \cdot \rho} = \frac1{\rho\cdot \ell}
$$
and there you have your answer: acceleration decreases with both density (so even small stones aren't noticably attracted) and with length (so small chunks of styrofoam are attracted more than big blocks of the same material). If this electrostatic acceleration is much less than the gravitational acceleration (always constant $\approx 9.81\:\mathrm{tfrac{m}{s^2}}$), it will look as if the object is not attracted at all. For instance, a big rock.
For stuff like straw, which mostly extends in a single direction, things are a bit different: here, changing the length scales electrostatic force quadratically and mass only linearly, hence bigger length actually leads to higher acceleration. (As long as the object is still further away from the charge than it is big.)
So,


*

*Styrofoam: attracted because of low density, provided the size isn't too big.

*Rocks, metal wire: not attracted because of high density.

*Straw, paper: attracted if length in a single direction is high enough to overcome the moderately-high density.



†Actually, it does depend substantially on material properties – conductance, dielectric number etc. – but length scaling is more important WRT to this question.
