We can disable captcha through Setup > General > Security Tabs will disable captcha globally.

But if do this,that will increase the spam support tickets.

Follow the stpes if you want to remove captcha only from Domain Checker page(www.yourdomain.com/whmcs//domainchecker.php)

Remove the following codes from domainchecker.tpl file.

Line number 3 to 7

{if $inccode}
<div class="alert alert-error textcenter">{$LANG.captchaverifyincorrect}</div>
{/if}

Find the following code (line number 28 to 37)

           {if $capatacha}
        <div class="captchainput">
            <p>{$LANG.captchaverify}</p>
            {if $capatacha eq "recaptcha"}
            <p>{$recapatchahtml}</p>
            {else}
            <p><img src="includes/verifyimage.php" align="middle" /> <input type="text" name="code" class="input-small" maxlength="5" /></p>
            {/if}
        </div>
        {/if}

Replace with the following

<input type="hidden" name="direct" value="true">