Nope, there is (almost) nothing we can do to make the Earth heavier!
All the manmade objects actually consist of raw materials which come
from somewhere on or in the Earth to begin with. All we can do is
rearrange the atoms we find around ourselves.
That having been said, there are some very, very tiny exceptions
to this rule. We have brought rocks back from the moon, and may someday
bring some back from Mars, who knows? Many many more rocks fall on the
Earth of their own accord, though (meteors and meteorites. Even if a
meteor burns up in the sky as it falls in, its atoms are included in
the Earth's total mass -- the ashes fall down). Models of the formation
of the Earth in the first place usually involve many collisions of
small objects sticking together.
The net effect of the space program, I am afraid, is to reduce the
total weight of the Earth, as we have sent spacecraft out there we do
not expect to ever get back.
I don't think we can affect the orbit of the Earth significantly
by sending spacecraft out. One concern, though, is that a big asteroid
may collide with the Earth, causing a lot of damage. I'd imagine the
damage (floods, earthquakes, etc) would be of concern starting with
much smaller asteroid impacts than ones that would noticeably affect
the Earth's orbit. In short, if something happened to change the
Earth's orbit, we'd probably have worse problems to worry about.
(republished on 07/19/06)