One side of the argument for 'keeping the metabolic rate up' with eating frequency implies that more frequent eating patterns increase the metabolic rate.
A meta-analysis conducted on eating frequency notes that "studies using whole-body calorimetry and doubly-labelled water to assess total 24 h energy expenditure find no difference between nibbling and gorging. Finally, with the exception of a single study, there is no evidence that weight loss on hypoenergetic regimens is altered by meal frequency. We conclude that any effects of meal pattern on the regulation of body weight are likely to be mediated through effects on the food intake side of the energy balance equation". A review article conducted assessing 179 abstracts (of which 10 studies were deemed relevant to assess meal frequency and weight loss interactions) found no significant relation between meal frequency and weight loss, albeit calling for more long-term evidence. These results are found in other review articles on the subject matter.
Various individual interventions that modify meal frequency while keeping calories static find that there is no difference in metabolic rate (24 hour energy expenditure) between the two groups and that there are no changes in weight loss at the end of the trial periods. When calories are dropped significantly, metabolic rate declines slightly but overall declines based on calories and not meal frequency.
One recently published paper actually finds the opposite, and that when comparing 3 meals against 14 meals over a period of 36 hours in a metabolic chamber in healthy males, that there were no significant differences in total energy expenditure and a slight increase in resting energy expenditure in the lower frequency group.
Not too many studies look at increased meal frequency and body weight gain, but the limited evidence at this moment (this section, and the epidemiology section later on) indicate that the seen weight gain is due to caloric intake rather than frequency.
The other side of the equation for the 'keep the metabolic fire stoked' implies that the metabolic rate can become depressed during periods of 'not eating'.
After 36 hours of fasting, an increase in metabolic rate is seen (and does not change further when measured at 72 hours). Adrenaline was found to be increased at 72 hours (but not 36) and when measured at 48 hours adrenaline seems to induce a larger amount of heat production (thermogenesis).
In nonobese humans, Alternate Day Fasting (not eating every other day) does not result in a decrease in metabolic rate after 22 days (when instructed to eat twice as much food on days where they can eat, to compensate).
Studies undertaken during the Ramadan also note an apparent lack of a difference in overall metabolic parameters between fasters and non-fasters. Although some studies (most notably those in unhealth persons) show limited health benefits with Ramadan fasting if food intake is kept relatively stable although it seems variable. While metabolic rate has not been investigated much per se, it doesn't seem to change to a significant degree.
Large scale survey research does tend to show a correlation between eating frequency and obesity, with the 'nibbling' approach inversely correlated with BMI (fat people seem to eat less often, thin people tend to eat more frequently). These studies do not look at muscle mass per se, but at BMI; there does seem to be a trend that more meals per day increases body weight and BMI. There is limited counter-evidence, and is confounded with high activity levels.
Additionally, the ISSN's stance on meal frequency notes multiple observational studies that do not suggest that eating frequency affects weight loss (on a fundamental level). Of interest are a few that suggest a relation, but the correlation is eliminated once confounding factors such as smoking, drinking, and stress are controlled for; indicating that they may be the causative factor(s).
In Sum; Survey research appears to show that there is a indirect relationship between meal frequency and weight gain which may be due to increased calories overall. A lesser meal frequency may be associated with lower BMI (at the same caloric level) due to exercise.
There is not too much evidence to suggest that meal frequency per se does anything good or bad for the metabolic rate, but that is just an epidemiological indicator of other habits which do influence metabolic rate and weight changes.
Higher frequency of meal consumption may be beneficial for preserving muscle tissue. When comparing 3 meals against 14 meals per day (an extreme case), it was found that despite the same amount of calories and no difference in metabolic rate that the low-frequency group had a higher protein oxidation rate (106.9±7.1 vs. 90.6±4.3 g/d) or 17% higher protein oxidation rates compared to 14 meals a day. However, an intervention in obese individuals noted that when there were four meals eaten daily that there are no differences in weight loss when consuming 80% of your casein at one meal relative to 'pulsing' whey in four meals at 25%, with the casein group outperforming the whey group in the final length of the trial on nitrogen retention. This latter study noted higher protein oxidation and synthesis rates with whey, but a trend towards nitrogen retention (muscle mass retention) with casein.
Theoretically possible that more meals daily improves nitrogen retention, but the one recent human study on the matter suggests that remaining in a post-prandial state is more important (which can be done with slower absorbing proteins or more frequency, or both)
One of the aforementioned studies did note better glycemic control, as assessed by glucose AUC, in the 3 meals daily group relative to 14 meals. This has been seen before when comparing 2 meals per day against 12, where the lower frequency appears to have better glycemic control.
Lower frequency meals (3) relative to higher frequency meals (14), when the overall daily calories are the same, appear to be more satiating and produce less hunger.