Every step in traditional image processing is semi-destructive. Bit-depth is really only significant in processing, though many are happy as clams to process in 24-bit. Working in 48-bit workspace provides much finer steps across gradients, and specially when making profound corrections, the more robust image will survive better. However, for viewing, as Kevgermany says, we are still very much in a 24-bit world.
For a good scan or well balanced image from a digital camera where corrections are minor, bit-depth has little impact. While the image is being processed, it is simply data in RAM and has no file format. That comes when you save it. Saving it with TIFF or high-quality JPEG does not alter the colour information. The red, green and blue data for each pixel is written to the file. In the case of the JPEG, any redundant information that can be perfectly reconstructed when the image is opened is discarded, resulting in a much smaller file than the TIFF.
TIFF and JPEG are two of the most versatile formats, for totally different reasons. TIFF will save at almost every bit depth from black&white with no greys at all for faxes and optical character recognition all the way to 48-bit images. One can embed a JPEG thumbnail in a TIFF for fast viewing in some programs. It can also contain multiple images in a single file.
JPEG rules the Internet. Any picture on my site that I want to protect from use, I compress until I can begin to see artifacts. I restrict the size to 640 x 480 or less, so if anyone tries to further process it and print it at a size that can be useful, the artifacts will ruin it. The term "lossy" sends some people into a panic - which is completely unwarranted. It has generated a great deal of FUD(Fear, Uncertainty & Doubt). Compression makes many things possible - the songs we listen to with pocket players and computers, and the video we watch are compressed. The great power of the JPEG format is that you have total control of the amount of compression. For optimum quality with optimum size, Photoshop's Save for Web lets you designate the size of file you want, and it will figure the best way to compress the image to provide top quality at that size.
On the last magazine shoots of my career, a glossy, hi-tech Brit magazine specified that I submit JPEG. One of my long time friends and colleagues is a well known shooter in the USA, who also writes and shoots for a top photography magazine as a stringer. He has a booming trade in fine art photography. He shoots JPEGs exclusively. Sees no reason to bother learning to process RAW. JPEG can be of great quality - even with substantial compression.
I will never save any photo under 100% quality in JPEG.
Pay no attention to Internet FUD or even what I am writing here. Work with a typical file one day when things are a bit slack, and see for yourself. Try various levels of compression, and then use that to set your own baselines.
Above all trust your eyes.
Much of the FUD comes from people who don't actually do photography, just draw conclusions from what they read. We are VISUAL
artists. We work with actual images. We and our viewers can clearly see when quality diminishes. Theory is not always supported by practice. Test using content you actually work with, and trust your eyes. Only that which can be clearly seen, matters.