This is a good question and actually fairly complicated.
This suggestion assumes that if a certain amount of something is good, more must be better. Is this true for food? Water? Medicines? Chlorine in your swimming pool? We know the greenhouse effect is good, we also know a rapid increase in it is not. All plants alive today have adapted to and evolved in a atmosphere with between 180 and 300ppm carbon dioxide.
That said there has been some research demonstrating that some plants grown in a very high CO2 environment will grow larger. Not all. Slightly more vigorous growth has been seen in Poison Ivy, of all things. However, other factors will likely limit or even reduce growth including drought and changing climate without time for plants to adapt. Additional heat helps some plants, hurts others. And ultimately plants seem to adapt to the carbon dioxide, returning to their previous structure. The following website provides a good overview of the pluses and minuses for plants in higher CO2 environments:
The problem with the quantity of photosynthesizes on the planet is not lack of CO2 but lack of plant life due to deforestation on earth. In addition, pollution of the oceans is reducing sources of sea photosynthesis.
Even if we had some miraculous growth burst in plant life, it could never begin to absorb the large amounts of CO2 we are adding to the atmosphere. Further, any gains would pale next to the negative consequences of global warming.