Logarithmic Weil height

It depends on what you mean by $|cdot|$, but probably no.

If by $|cdot|$ you mean the absolute value on $mathbb C$, and your algebraic integers are elements of $mathbb C$, then the answer is no. The logarithmic Weil height of $1-sqrt3inmathbb A^1(overline{mathbb Q})$ is $frac12log(2)$, which is strictly bigger than $log(max(1,|1-sqrt3|))=0$. (To show that the logarithmic height is $frac12log(2)$, use the formula for the logarithmic height on $mathbb A^1(mathbb Z[sqrt3])$ below.)

However, a similar-looking inequality is true if we take account of all possible archimedean norms on the algebraic integers. Here is a precise statement. Let $a_1,dots,a_n$ be algebraic integers in $mathbb C$. Then $$ h(a_1,dots,a_n)leqmax_{1leq ileq n}left(max_{sigmain G_{mathbb Q}}log(|sigma(a_i)|)right),, $$ where $h$ denotes the logarithmic Weil height on $mathbb A^n(K)$ and the second $max$ is taken over all field automorphisms of $overline{mathbb Q}$.

This follows relatively straightforwardly from the definition of the Weil height. To cut a long story short, if $K$ is a number field of degree $d$ over $mathbb Q$, then the Weil height on $mathbb A^n(mathcal O_K)$ is given by $$ h(a_1,ldots,a_n)=frac1dsum_{sigmacolon Khookrightarrowmathbb C}log(max_{1leq ileq n}(|sigma(a_i)|)) ,, $$ where the sum is taken over all complex embeddings $sigmacolon Khookrightarrowmathbb C$. This is just what you get by specialising the usual formula for the Weil height on $mathbb P^n(K)$. It follows from this that $$ h(a_1,dots,a_n) leq max_{sigmacolon Khookrightarrowmathbb C}max_{1leq ileq n}log|sigma(a_i)| ,, $$ which implies the claimed inequality.

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Remark: This all depends, of course, on the chosen normalisation of the Weil height. Since the OP was talking about "algebraic integers", I presumed they were considering Weil heights with the normalisation which leads to a Weil height on $mathbb P^n(overline{mathbb Q})$.