It's possible but not as likely as in Europe, at least after the Tang dynasty. When the Tang ended in 906 AD the following turmoil during the Five Dynasties and Ten Kingdoms period saw the end of most of the pre-existing elite genealogies that dated all the way back to the warring states or prior era whether fictitiously or otherwise. Genealogies took a backseat in importance during the Song dynasty and were eclipsed in status by official office titles in the government. This is not to say prestigious families or clans, however you call them, did not exist, but they were based primarily on a narrative of prestigious offices once held by an ancestor.

Chinese dynasties were also not officially monogamous in the way Christian European dynasties aspired to be. A cursory look at any Chinese ruler's wiki will give you a list of all their concubines in addition to their wife, who was often interchangeable depending on whom the emperor favored at the moment and was subject to change. Despite this, in reality most Chinese families even prior to the modern era were nuclear with children and their grandparents as dependencies. Concubines were common for those who could afford them but this was probably a minority.

Chinese families also practiced a type of exogamy where sons or daughters were not allowed to marry someone of the same family name. So Zhaos avoided marrying Zhaos and Lis avoided marrying Lis. This would have excluded very large portions of the population at any time post-warring states, as Chinese surnames had mostly been whittled down to a hundred. As early as the Sui dynasty prestigious surnames had already become common enough that they no longer held the value that they once did, and clans of the surname differentiated among each other using choronyms. For example, a very famous family might call themselves the Zhaos of Zhengling or Zhengling Zhaos, Xiangyang Lis, Dongfeng Wangs, etc.

These clans were large and many of their members never even set foot in their clan estates. The Chinese dynastic capitals usually held a hegemonic role in the lives of elite families, certainly Chang'an did, and important members often chose to live at the center of politics rather than remain in their traditional locales. Many abandoned tending to their properties altogether and relocated to Chang'an. This was part of reason why the late Tang rebellions such as Huang Chao and other disasters which damaged Chang'an led to the downfall of the elite lineages that had tied their fate to the court. According to David Johnson, the end of the Tang saw "class wars" fueled by "violently anti-aristocratic feeling" and that by the end of the 9th century, the aristocracy had become an ideal rather than a social or political reality. A new, more diffuse collective of elite families who justified their dominance over society and politics on the basis of merit, talent, and education would arise during the Song. This "Confucian" literati elite is the one most people are familiar with when thinking of China.

Epitaph of Liu Neizhe (822):

"When ranking [clans] from around the empire, there are distinctions between those who lead and those who follow; when assessing them, it is unambiguous which are important and which are not. There are those [individuals] who receive the teachings of righteousness and harmony, and who respect the importance of inherited achievements. These individuals have deep esteem for their ancestries and do not take lightly their family relations. Thus, they inevitably marry people of their own station: the men are granted fitting consorts, and the women are matched to [husbands] to whom they submit. With law and morality in perfect agreement, distant and near kin are not at odds with each other. The caps and sashes [of their elegant clothing] are displayed with profusion; their descendants prosper and flourish. This is the way to preserve one's family while respecting one's origins. [One can, thus,] hope never to decline for one hundred generations. Then there are those [individuals] who abandon their families and forget their origins, who go against their duties in order to pursue their own interests. They debase their ancestries while esteeming other [lesser] families; they value wealth, while breaking the [proper] order of the ranks. Thus, they inevitably associate themselves with those who are [socially] remote and distance themselves from those who are [socially] akin. They look towards their own livelihood without understanding the virtue of proper marriage ties; they look for shortcuts without understanding the [more onerous] path of righteousness. This is called the dilution of one's customs and the severe tarnishing of one's family heritage." 國朝差敘，則先後有別：品藻，則經重甚明。其有本仁義雍和之教，稟閥閱相承之重，深敬祖始，不忘吾耦。則必幕族類而婚，依族類而嫁，使男得其配，女適其歸，法教無二途，疏戚無閒言。纓緌紛綸，枝葉蕃昌。是為克家敬本知道，冀不失於百代也。其有捨族忘本，異尚封己，卑祖使而尊他們，厚幣財而分甲乙，則必親其所疏，疏其所親。顧衣食而不知配耦之端，視步武而入知仁義之塗。其風俗之澆薄，保家之甚病。

Song Scholar Zheng Qiao (1104-62):

"Up until the Sui and Tang dynasties, officials had dossiers [identifying the offices of their ancestors], and families had genealogies. The appointment of officials relied upon the dossiers; marriages between families relied upon genealogies... Ever since the Five Dynasties, one no longer asks about family background when selecting officials, and one no longer asks about family prestige when arranging marriages." 自隨唐而上，官有簿狀，家有譜系，官之選舉，必由於簿狀，家之婚姻，必由於譜系… 自五季以來，取士不問家世，婚姻不問閥閱。

So at least from what I have read, at no time has the idea of an incestual genetic heritage been mentioned in Chinese history books, except as a taboo, slander or insult. Nor do genetic disorders feature very prominently in dynastic histories if at all, however it's not impossible that they were intentionally omitted out of fear of reprisal. There are instances of successors being chosen specifically for having survived epidemics, such as smallpox, but none that are genetic. This wouldn't be too strange given the three reasons I've listed: the lack of an entrenched class of circumscribed noble bloodlines aside from the royal family following the Tang, polygamous relationships, and exogamous marriages. It could also be an oversight in historiographic tradition, and the Chinese simply never focused on it, but given that all Chinese emperors were polygamous, offspring of various descents were numerous and securing a healthy successor does not seem to have posed too much of a problem.

Sources:

Chinese History: A New Manual by Endymion Wilkinson

The Destruction of the Medieval Chinese Aristocracy by Nicolas Tackett

Just to add to this, can anyone recommend some reading on the subject of medieval lineages and their genetic disorders?

I have a similar question about the Goryeo dynasty in Korea, since according to what I know about it many kings ended up marrying their half sisters. Do we have any historical evidence that genetic diseases presented in the royal family during that time period?

Sorry if I'm breaking rules but why do you assume it being taboo is the only thing keeping people from doing incest? Incest in European royalty is a phenomenon, not some inevitability.

How inbred were European kings and aristocrats?

Could OP provide some examples of incest leading to genetic disorders amongst European kings and aristocrats?

Dumb question, but why were Concubines an expensive hobby?

I understand why they're, but I figure the regular Mongolian in some raid obtaining some slave/concubines, or is that an example to foreign?