$\DeclareMathOperator{\Sp}{\mathrm{Sp}}$I am taking a special case $\Sp$ here, mainly because it has nice categorical properties. 

Let $R$ be an $E_\infty$-ring spectrum. In [Higher Algebra][1], Lurie proves we have a forgetful functor (part of monadic adjunction)
$$ U_R:\operatorname{Mod}_R(\Sp) \rightarrow \Sp$$
where $\Sp$ is in the $\infty$-category of spectra. 

$U_R$ reflects equivalences. But **is $U_R$ faithful** in the sense that that the induced map of 
$$Map(x,y)\rightarrow Map(U_Rx,U_Ry)$$
 mapping spaces is $0$-truncated in the $\infty$-category of spaces. i.e. the homotopy fibers are [$0$-truncated][2]. 


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One categorically, $U$ is faithful in many cases, i.e. if we replace $\Sp$ with $\mathrm{Ab}$. 
Perhaps the answer is false in $\infty$-categories. 
 I'd like to understand what goes wrong. Some comments on the following would be helpful: 

 - A counter example where $U_R$ is not faithful. (i.e. is it faithful when $R=H\Bbb Z$? )
- A brief/reference explanation for what accounts of this. 


  [1]: http://people.math.harvard.edu/~lurie/papers/HA.pdf
  [2]: https://ncatlab.org/nlab/show/n-truncated+object+of+an+%28infinity%2C1%29-category