UnicodeIt

Tested on Linux, Mac and Windows:

Converts LaTeX tags to unicode. Available online at unicodeit.net.

Examples

\alpha α, \beta β, \infty ∞       e^+ e⁺, \mu^- μ⁻               \exists ∃, \nexists ∄
\int ∫, \sum ∑, \partial ∂        \to →, p\bar{p} pp̅             \mathcal{H} ℋ, \mathbb{R} ℝ
\slash{\partial} ∂̸                \underline{x} x̲                \phone ☎, \checkmark ✓
\dot{x} ẋ, \ddot{x} ẍ             A^6 A⁶, m_0 m₀                 \Im ℑ, \Re ℜ, \hbar ℏ
\gamma γ, \Gamma Γ                \~{O} Õ                        \perp ⊥, \parallel ∥
\sfrac{3}{5} ⅗                    \therefore ∴, \because ∵       \subset ⊂, \supset ⊃

Python

Install with pip install unicodeit and run

python -m unicodeit.cli \\alpha

or in Python

import unicodeit
print(unicodeit.replace('\\alpha'))

JavaScript / TypeScript

Install with npm install unicodeit --save-dev and use it like this:

var unicodeit = require('unicodeit');
console.log(unicodeit.replace('\\alpha'));

Mac Automator

Create your own Automator Workflow:

Create a new "Quick Action" (might also be labeled as "Service").
At the top, leave the defaults: "Service receives selected text in any application"
Select the checkmark "output replaces selected text".
Add the action "Run Shell Script".
From dropdown, select to "pass in: as arguments".
The command is: /usr/local/bin/python3 -m unicodeit.cli $1. This Python interpreter must have unicodeit installed; e.g. with /usr/local/bin/python3 -m pip install unicodeit.
It should look something like this:

To set a keyboard shortcut, go to System Preferences → Keyboard → Shortcuts → Services → Text → UnicodeItAction. Choose a keyboard shortcut like Command+Option+Shift U.

Now you are all set to use your new keyboard shortcut in many apps, like here in Keynote: