One of the possible drawbacks of modernist art, depending on one's perspective, is that it borrowed heavily from West African influences in sculpture but took little interest in the cultures from which the artifacts were derived.
In Paris, West African masks and sculptures were sought after as early as 1906 but became particular objects of interest during the period between the wars, collected by the American expatriate writer Gertrude Stein and the gallery owner Paul Guillaume. African art also had a substantial influence on the development of Cubism, particularly as it was envisioned by Pablo Picasso.
Many European critics at the time, particularly French critics, believed that European culture could rejuvenate itself through a superficial exploration of African art and cultures. I write "superficial" because Europeans expressed little interest in learning about African cultures as they actually were; instead, they superimposed their own ideas about the region and its peoples, which were often very stereotypical. They believed, for example, that African culture amounted to little more than primeval expressions of sexuality and sensuality, as well as more primitive forms, and sought to superimpose that in European art. The interest in African art was a rejection of the machine age. Machinery was integral, after all, in the destructive war that had broken the European spirit. In France, where the interest in African art was particularly pronounced, the nation suffered massive losses in World War I.
So, I would argue that one of the drawbacks of modernism was that it fetishized African, as well as Black American, cultures with the misguided belief that these cultures could rejuvenate artistic production and culture in Europe. On the other hand, without these influences, we would not have Cubism, and jazz would never have made it to Europe, especially France, where it retains cultural influence.
Although modernism was an important movement in the arts and resulted in many outstanding works, there are several possible criticisms of it.
First, modernism (and postmodernism, to an even greater degree) is essentially parasitic on earlier traditions. Breaking the boundaries of meter, for example, only has meaning if regular meter is a standard poetic convention. Once a convention disappears, fighting against it is meaningless; this means that modernist works can appear dated once their project succeeds. Many aspects of modernism embody ironic re-appropriation of Victorian material, something only relevant at a time when Victorian standards were commonly accepted.
Next, modernist art is inherently elitist. It is highly allusive, hermetic, and cerebral, appealing only to a limited audience. This tends to weaken public support for the arts. Taken with the standard mode of modernism being critique, artistic modernism can end up in unintended de facto opposition to the arts.
At times, modernist works in their explicit self-reflexivity become almost a commentary on art, so that what the reader experiences in not so much the sense of discovering a beautiful poem or painting but of reading a scholarly journal.
Modernism was an important aesthetic and cultural movement that is generally thought to have begun in the late 19th century and to have remained extremely influential at least until the WWII era. Though it is an important historical period, especially in terms of the art it produced, the Modernist era also wrestled with some seriously negative ideas, and these could be seen as potential drawbacks. For instance, Modernist literature often explores the lives of individuals that have been alienated from society, or who feel isolated from the culture they live in. Moreover, many Modernist writers tended to suggest that, since many traditional values were destroyed by advancements in science, philosophy, and more, existence was meaningless and lacked any kind of solid foundation. Now, it's important understand that these were and are important concepts to explore. That said, these ideas could also be seen as the drawbacks of Modernism, as they advance a thoroughly pessimistic and depressing view of life.