When going from cup size 30 H to 30 C, how much breast tissue, in weight, is removed?

Thank you for your question. 
Unfortunately, it is impossible to exactly answer this question for several reasons. 
The measurement of cup sizes varies from store to store, and from bra manufacture to bra manufacture.   If a woman is measured a D cup in one store, there is a chance of her being measured a C, DD, or even a B cup in another store. 
Breast tissue weighs differently from woman to woman. It depends on how dense the breast is. Younger woman often have very dense breast tissue, while older woman have more fatty breast tissue, which in comparison weighs less. So the same amount of breast tissue removed in 2 different women, may weigh a different amount. 
When I speak with my patients about breast reduction surgery, I tell them that I cant guarantee a certain cup size. I asked them what size they would like to be if they could choose, to get a good understanding on how the woman would like to look after breast reduction surgery. But the insurance company does require a certain amount of breast tissue to be removed (based on height and weight of the patient) in order to qualify for surgery. My goal with surgery is, to follow the insurance guidelines, but to still leave the woman with a nice , aesthetically pleasing, perky breast shape matching her body shape and as close to the breast size the woman would like to be. 
Best wishes, 
Franziska Huettner

Breast reduction eligibility and coverage by insurance varies from one insurance carrier to the next, and cannot be determined based on cup size, but only based on an in-person examination to figure out how much tissue the surgeon needs to remove.  You also often have to qualify for multiple other criteria from your insurance carrier, so be sure to request their breast reduction coverage policy ahead of time before your plastic surgery visit to ensure you meet all the other non-size related demands that they have.  Be sure to discuss your goals and concerns with your board certified plastic surgeon and consider the pros and the cons. Be certain your surgeon is properly certified.  I've attached a link where you can start your search.

It is really not possible to exactly quantify how much weight will be removed, as every manufacturers cup sizes differ, so this is not a good metric to use.  More than likely you will end up larger than a C-cup, as it is also very difficult to maintain breast tissue flap and nipple-areolar viability if attempting to remove enough tissue to go from an H-cup to a C-cup.
 After reviewing your description, it seems that that you would be a great candidate to have a breast reduction and lift using the Bellesoma technique.   

Bellesoma is a new breast reduction and lift technique that seems like it would be a great fit for you. The Bellesoma procedure reduces the sagging and weight of your breast, lifts your breast and reduces areola diameter, preserves the nipple sensation, and looks as full and perky as if an implant had been placed without using any implant, so no need to compromise.  One of the huge positives of this technique is that there is no vertical incision traveling down the front of your breasts.
Some of the differences between Bellesoma and the traditional breast reduction/lift techniques:
-Bellesoma creates upper pole fullness that traditionally needed an implant to achieve.
-There is NO vertical scar, only well-hidden scars in the fold under the breast, and around the areola.
-The breast is dynamically supported by attaching it to the muscle, so there is no recurrence of sagging which are expected with traditional lifts or reductions.
-The entire central portion of the breast is undisturbed, so nipple sensation remains intact and is very unlikely to be compromised.
-For absolute best results, patients should have a stable weight and have a BMI under 30.
The procedure is accomplished by using the skin of the lower pole of the breast (the part below the nipple that sits in the bra cup) to shape the whole breast into a perky dome, then straps are made of the extra skin, anchoring it to the underlying chest muscle so that there is virtually no chance of repeat sagging.  The skin that above your nipple and below your collar bone is utilized to cover the perky, lifted dome that has been created and then a new (usually smaller) circular opening for the areola is placed at the high point of the cone, creating your new lifted, full and shapely breast.
From your description, you seem to be a very good candidate for this new technique and should really get the look that you are desiring without implants or unnecessary visible scars. I would recommend that you search for an ABPS board certified plastic surgeon who is also a member of the American Society of Plastic Surgeons and the American Society of Aesthetic Plastic Surgeons in your area or an area that you would like to travel to on holiday who offers the Bellesoma technique to discuss your breast reduction and possible results.
Best wishes!
Dr. Brown