A low-carb, high fiber diet, is the "natural" human diet. For the two million years before agriculture was developed, humans lived as hunter/gatherers. The (plant) fiber content varied according to the seasons, and the environment, but nutritional anthropology tells us, that they ate a lot more fiber than we do (The Inuit of Alaska, Canada, Greenland, Siberia.... being the notable exception. Some argue that they can survive on a low fiber diet, only because of the unique nutritional properties and the large amount of fat that is present in the cold-water sea animals that make up so much of their diet).
If you only ate celery, or woodpulp I suppose it would be possible, that your metabolism could be impacted, but if you're eating a wide range of plant foods, this isn't going to be a problem (vegans and other vegetarians do so every day).
Some scientists (the nutritional anthropologiest and other nutrition scientists) estimate that humans are designed to eat 100 to 150 grams of fiber, which makes the 30 to 35g recommendation of the current governmental "healthy guidelines" seem paltry.
In addition, fruits and vegetables have many other micronutrients that are important to a healthy diet.
If you ate only fiber (cellulose - wood pulp or sawedust, essentially) you could have a problem in theory, but that's not how humans get fiber. We get it from fruits, vegetables, and grains - from which there are plenty of nutrients and calories to fully sustain us. Which is why weight loss is not a guaranteed outcome of a high fiber diet. You can still eat enough calories to maintain or even gain weight - if you don't watch portions.
When I first started following South Beach (a high fiber, "good carb" diet), I lost weight initially, because the higher fiber foods meant I felt like I was eating the same amount that I was to maintain my weight. Because I was actually eating fewer calories for the same amount of volume, I lost weight - until I didn't. My failure to continue to lose, wasn't starvation mode - it was portion shift. I thought I could eat more and still lose, and it worked for a while. But overeating fruits and vegetables and whole grains, is still overeating. Portion control is still important.
As Sarah pointed out, suddenly increasing fiber can cause digestive troubles (to put it mildly, the actual experience can make you wish for merciful death. It can be that painful, especially if you have IBS).
There are other benefits to a high fiber diet, such as a tool in the prevention of digestive system cancers, particularly colon cancer. A high fiber diet is associated with other health advantages - resistance to heart disease, cholesterol issues, diabetes.....
Adding fiber is a good thing, not only for weight loss but for overall health (thought o be clear, I mean natural fiber, from fruits and vegetables - I'm not talking about using fiber supplements such as Metamucil as your primary source of fiber)