In theory it's possible because it needs approval by the European Council.
In the Treaty on the European Union in Article 16(9) you'll find:
Article 16
(...)
- The Presidency of Council configurations, other than that of Foreign Affairs, shall be held by Member State representatives in the
Council on the basis of equal rotation, in accordance with the
conditions established in accordance with Article 236 of the Treaty on
the Functioning of the European Union.
In Article 236 you'll find:
Article 236
The European Council shall adopt by a qualified majority:
(a) a decision establishing the list of Council configurations, other
than those of the General Affairs Council and of the Foreign Affairs
Council, in accordance with Article 16(6) of the Treaty on European
Union;
(b) a decision on the Presidency of Council configurations, other than
that of Foreign Affairs, in accordance with Article 16(9) of the
Treaty on European Union.
That being said notice that some members have already stepped aside voluntarily from the Presidency of the Council of the European Union. The most recent was the UK in 2017 (source). The linked Wikipedia page (to the presidency) also mentions Germany switching places with Finland at some point. So it's not unheard of members stepping aside (even if temporarily). However a move by the Council to remove the Presidency of a member (which I consider unlikely) would be, I believe, something new. I consider much more likely that closed discussions would take place to convince Romania to swap its Presidency slot for some other in the future (or something equivalent) and in the process save face for all involved.
EDIT: This EU Observer (small) article came out just a couple of hours ago (14, November, 2018). Notice that Finland is supposed to take up the Presidency slot after Romania.
Finnish prime minister Juha Sipila has told the country's STT news
agency that Helsinki was making preparations to take over the next EU
presidency, starting 1 January, from Romania, but that Bucharest had
not filed a request to do so. He spoke after Romanian president Klaus
Iohannis said his country was "totally unprepared". Romanian PM
Viorica Dancila told press Tuesday: "I assure you that Romania is
ready".
EDIT2: (15, November, 2018) Things are looking bad. Yesterday a speaker for Romania ruling party reacted aggressively towards the complaints made by EU about the judicial reforms in Romania. Similarly to what happened in Hungary and Poland, the EU is concerned that the reforms will erode an independent judiciary. To make matters worse the first rumors of a "Romexit" have started.
Romexit?
"It's surprising that a country that had so many benefits
from the EU ... [would] kick all of these opportunities away," Corina
Cretu, Romania's EU commissioner, who is in charge of regional funds,
billions of euros of which have flowed to Bucharest since it joined in
2007, said on Wednesday.
"I don't think anybody wants this and I'm sorry that the possibility
that Romania would ever leave the EU has come, I don't know, out of
nowhere. This, in my opinion, would be a national catastrophe," she
told Romanian TV.
She urged the PSD party, to which she herself belongs, not to see the
EU action as "a conspiracy, as if someone wanted to hurt us".
"It is a result of what has been happening [in Romania] lately," she
said.
