One important point to keep in mind is that apartment sizes in Mumbai may be smaller that the ones in Delhi for the same level of rent. So if you have bulky furniture in your apartment in Delhi, it may or may not fit in your Mumbai apartment. You will probably be able to take this decision better once you have identified the apartment in Mumbai.
As Sonali mentioned, Mumbai apartments are much smaller than Delhi apartments – so size is one factor
Furniture that you may have bought in Delhi, may work well for the extreme hot and cold temperatures, but Mumbai, depending on where you finally choose an apartment, you may need furniture that is resistant to fungus / mold because of the sea breeze – this is not a problem with modern furniture, but some ancestral heirlooms could be totally ruined by the salt content in the breeze, depending on how close you are to the sea.
Whether to rent furnished or unfurnished, depends on
1. what furniture you already have and are planning to carry with you.
If you aren’t carrying your existing furniture with you, where will you leave it behind?
Will you be paying more for storage by not carrying it with you?
Are you renting out your place in Delhi as a furnished space?
Are you worried about wear & tear of your furniture if it is left behind?
2. What is your lifestyle like?
Do you own minimal stuff, so its easy for you to move?
Do you stay asset light and travel with just your personal items?
In this case, you may prefer to look for a furnished apartment, it will save a lot of hassle
3. Are you starting from scratch with no furniture of your own?
Do you plan to buy furniture as an asset that you will treasure or do you want something that can be easily resold when you leave
This option is very time and energy intensive – to buy all your furniture from scratch
Depending on your long term plans, a furnished apartment may be an easier option