Growth is a pretty complicated process, and requires a lot of energy. All living things grow, and thus all living things need energy for growth (and for a lot of other things, but growth is a biggie). As I'm sure you know, animals get their energy from food: we eat organic carbon (carbohydrates, proteins, fats) and through a process known as the Krebs cycle get energy by converting the organic carbon to inorganic carbon (carbon dioxide, CO₂). Plants also get their energy for growth from the Krebs cycle. But as I'm sure you know as well, plants don't eat (with the exception of a few strange plants like the Venus flytrap). So how do plants get the energy they need for growth? They harness it from sunlight in a process known as photosynthesis.

Photosynthesis is sort of the reverse of the Krebs cycle: it stores energy by converting inorganic carbon (carbon dioxide) into organic carbon (glucose, for example). This stored energy (organic carbon) enters the Krebs cycle and this is how the plants get the energy they need for growth and survival. So the answer to your question is no, plants can not grow or survive without photosynthesis.

Plant growth, and certainly plant survival, may occur without photosynthesis over very short periods. (Be aware that growth and survival are different: growth implies an increase in size or weight, so things can survive and not grow.) If you stick a fairly large plant in the dark, it won't die immediately. It may even grow a little bit, if it has some stored energy. Just like if you ate the biggest meal of your life and then stopped eating immediately, you might still put on weight initially from the huge meal (you'd grow), but eventually you would start to loose weight, and finally you'd starve to death. So the plant might grow a bit in the dark, without photosynthesis, from stored reserves, and might continue to survive for a couple weeks or more without growing, but in the long term the plant will not be able to grow or live without photosynthesis and would eventually die.