Wouldn't radiolabelled phosphorus in DNA break it apart as it disintegrates?
Of course it would break, just like you said; also, a high-energy $\beta$ particle would kill quite a lot of bystander molecules. Also, if not for other reason, the resulting molecule would no longer be DNA , since the decayed atom would no longer be $\rm P$. Also, the product would no longer be radioactive, so we wouldn't be able to detect it anyway.

The same applies to any other case of radiolabeling, in biochemistry or elsewhere, and you seem to be missing the very point of it: All this is not important.

Say, we have some $\ce{A + B -&gt; C + D}$, and we label some atom in $\ce{A}$ so as to find out whether it will go to $\ce{C}$ or to $\ce{D}$. Then we run the reaction as usual. Now, radioactive atoms do not decay all at once; some decay earlier and some later. Those which decay earlier (that is, before and during the reaction) are lost to us, and we don't care about them one bit. Then we separate $\ce{C}$ from $\ce{D}$ and measure the radioactivity of each, and it is those atoms which decay now that matter.

Oh, and it is not like we label all atoms in $\ce{A}$. No, a very tiny fraction would suffice. Those viruses would probably get but a few radioactive atoms each. Some will be lucky enough to reproduce. Some will be killed by an early decay and remembered as martyrs of science.

So it goes.