No. The grounds of inadmissibility cited in the article do not apply to heads of state or heads of government, nor to official travel by diplomatic personnel or officers or employees of international organizations. (Such people can be refused entry by diplomatic means, for any reason or for no reason at all, but that does not depend on past drug use.)
The lawyer quoted in the article is correct as far as Gove's personal travel to the US while he is not prime minister. However, the article sets the context by mentioning "speculation that if he were to become the next British prime minister, he could be banned from visiting the US."
For travel while he is prime minister, or for official travel in any other capacity, the cited grounds of ineligibility do not apply, because of section 102 of the Immigration and Nationality Act (INA), codified at 8 USC 1102:
Except as otherwise provided in this chapter, for so long as they continue in the nonimmigrant classes enumerated in this section, the provisions of this chapter relating to ineligibility to receive visas and the removal of aliens shall not be construed to apply to nonimmigrants—
(1) within the class described in paragraph (15)(A)(i) of section 1101(a) of this title, except those provisions relating to reasonable requirements of passports and visas as a means of identification and documentation necessary to establish their qualifications under such paragraph (15)(A)(i), and, under such rules and regulations as the President may deem to be necessary, the provisions of subparagraphs (A) through (C) of section 1182(a)(3) of this title;
(2) within the class described in paragraph (15)(G)(i) of section 1101(a) of this title, except those provisions relating to reasonable requirements of passports and visas as a means of identification and documentation necessary to establish their qualifications under such paragraph (15)(G)(i), and the provisions of subparagraphs (A) through (C) of section 1182(a)(3) of this title; and
(3) within the classes described in paragraphs (15)(A)(ii), (15)(G)(ii), (15)(G)(iii), or (15)(G)(iv) of section 1101(a) of this title, except those provisions relating to reasonable requirements of passports and visas as a means of identification and documentation necessary to establish their qualifications under such paragraphs, and the provisions of subparagraphs (A) through (C) of section 1182(a)(3) of this title.
The correspondence between visa categories and the statute is given at 8 CFR 214.1(a)(2).
A prime minister, as head of government, is in class A-1 regardless of the purpose of travel (see 9 FAM 402.3-5(C)(1)) and therefore falls under 8 USC 1102(1), quoted above. Any official travel to the US in a capacity other than as prime minister will be covered either by category A-1 or A-2, so will fall under either paragraph 1 or 3 quoted above.
In both cases, the only grounds of inadmissibility that can apply are those in 8 USC 1182(a)(3)(A), 1182(a)(3)(B), and 1182(a)(3)(C). But the grounds mentioned by the lawyer (see the linked page, and keep in mind that INA section 212 corresponds to the US Code, Title 8, section 1182) are 1182(a)(2)(A)(i)(II), 1182(a)(1)(A)(iii), and 1182(a)(1)(A)(iv).
In conclusion, the "speculation that if he were to become the next British prime minister, he could be banned from visiting the US" is backward. His term as prime minister, should he ever achieve that office, would be the only time during which he could not be found inadmissible to the US because of his admission.