Some observers and vendors, especially vendors of small telescopes, claim that large apertures are useless or even counter-productive under heavy light pollution. In my opinion, and the opinion of most experienced observers, there is essentially no truth to this claim. If anything, satisfactory deep-sky observing requires more aperture under heavy light pollution than under dark skies.
However, large apertures are not a substitute for dark skies, any more than dark skies are a substitute for large apertures. Small apertures and bright skies both hurt deep-sky objects, but they hurt them in fundamentally different ways. The reason lies in the distinction between an object’s integrated brightness — the sum of all the light that comes from it — and its surface brightness, or brightness per unit surface area.
The galaxy M33 is one of the brightest of all the Messier objects as measured by integrated brightness; however, that light is spread out over an area considerably larger than the full Moon, making it one of the faintest of all the Messier objects as measured by surface brightness. The tiny planetary nebula M76 is about fifty times fainter than M33 as measured by integrated brightness, but its area is only about one thousandth that of M33, giving it a surface brightness twenty times higher. This subject is explored in more detail in the section Surface Brightness.
Small aperture places a lower limit on the integrated brightness of the targets that can be observed, while skyglow places a lower limit on the surface brightness. Thus, M33 is faintly visible to the unaided eye in dark skies, and very easy in small binoculars, but it is extremely hard to see from the city in any telescope, no matter how big. M76, by contrast, is a bit of a challenge in my 70mm scope even under the darkest skies, but it is quite easy and attractive in my 178mm scope even from the city. In fact, it shows nearly as well in that scope from the city as it does under dark skies; its surface brightness is high enough to overpower the urban skyglow.
Stars and star clusters are the objects where aperture and sky brightness are most nearly interchangeable. My 178mm Dob under urban skies sees very nearly the same stars as my 70mm refractor under fully dark skies; therefore, a star cluster looks much the same through the bigger scope under urban skies as through the smaller scope under dark skies. But even that isn’t completely true, because getting an optimal view in the bigger scope requires higher magnification, which is likely to frame the cluster poorly, and make it stand out less well from the background. Moreover, many clusters, especially globular clusters, are only partially resolved even in the biggest telescope, and the unresolved stars merge to form a nebulous background which is hurt severely by light pollution.
Bright planetary nebulae like M76 and M57 are the objects where aperture matters most and sky brightness least. These objects are like their namesakes the planets; they easily outshine even the brightest skies, but require high magnification which can be provided only by large apertures. But even bright planetary nebulae often have faint halos that can be obliterated by light pollution.
For galaxies, dark skies are of paramount importance. With a handful of exceptions, all of the Messier galaxies show far better in my 70mm refractor under dark skies than in my 178mm Dob under urban skies, or often even under suburban skies. Our own Milky Way galaxy is a good indicator. If the skyglow is bright enough to swamp the Milky Way, making it invisible to the naked eye, then it will also swamp the outer disks of most galaxies as seen through a telescope, and you will be reduced to viewing the nearly featureless cores. If the Milky Way is visible but unexciting to the naked eye, so will most other galaxies be through a telescope. If Milky Way glistens with detail, so will most other galaxies.
The links below show pictures of various Messier objects through both my 70mm telescope and my 178mm telescope under urban and suburban skies. For the prototypic galaxy M31, even the fairly modest difference in sky glow between the two sites is far more important the aperture. For the star cluster M38, aperture is far more important. The situation is more complex with respect to M42, the Orion Nebula. This objects has aspects with high surface brightness that benefit most from aperture and also aspects with low surface brightness that are affected more by sky glow.