Specially for preparation metal oxide nanoparticles using co-precipitation method. If crucible is open, then there may be risk of contamination inside the muffle furnace. I want to clarify this point.
Yes having a closed lid would help you not only decrease the contamination but also decrease the amount of carbon dioxide that might be incorporated within your particles.
Some compounds however like hydroxyapatite can be prepared in an open mode as they sometimes need to be prepared in their carbonated form. So your choice would depend on whether you would need the presence of carbon dioxide in your compound or not.