The results of this project can be stunning or commonplace, it just depends on what kind of bottle is used.
Note: these glasses are not recommended for use with hot liquids. The plastic is a thermoplastic, which means that it can deform, melt, or release chemicals or particles into the air or your drink. Please use them only with cold drinks. These are also alcohol-safe.
Extensive research has been done on the reuse of plastic bottles. This research has found that PETE (Plastic number 1) bottles do not pose a health risk to be reused, as long as they were originally intended for food products, and are in good clean condition. Just don't put hot liquids in them or leave them in the sun for extended periods, as the solar radiation can break down the plastic. Any leaching of chemicals into contained liquids has been found to be negligible, far below the levels of water quality safety standards. Furthermore, all these tests were conducted on liquids that had been in the bottle for an extended period, and since the outcome of this project is a cup of one sort or another, the liquid probably will only be in the cup for a few hours max. Sodas, soft drinks, and alcohols are all sold in PETE plastic bottles, so there is no risk in using them again. Once the cup has been in solid use for a year or so it might be a good idea to replace it, but normal use is not an issue.
Tl;Dr: it is safe to reuse PETE plastic bottles because science says so.