Core-547 should help deal with this by allowing to hide/require certain fields, but in the mean time...this should work for you.
In /plugins/order/views/templates/standard/types/registration/signup.pdt
Find:
<hr>
<div class="row panel-fix">
<div class="col-md-12">
<h4><?php $this->_("Signup.index.heading_billing");?></h4>
<div class="form-group">
<?php
$this->Form->label($this->_("Signup.index.field_taxid", true), "tax_id");
$this->Form->fieldText("tax_id", $this->Html->ifSet($vars->tax_id), array('id' => "tax_id", 'class' => "form-control", 'placeholder' => $this->_("Signup.index.field_taxid", true)));
?>
</div>
</div>
</div>
Replace with:
<!-- <hr>
<div class="row panel-fix">
<div class="col-md-12">
<h4><?php $this->_("Signup.index.heading_billing");?></h4>
<div class="form-group">
<?php
$this->Form->label($this->_("Signup.index.field_taxid", true), "tax_id");
$this->Form->fieldText("tax_id", $this->Html->ifSet($vars->tax_id), array('id' => "tax_id", 'class' => "form-control", 'placeholder' => $this->_("Signup.index.field_taxid", true)));
?>
</div>
</div>
</div> -->
<?php
$this->Form->fieldHidden("tax_id", $this->Html->ifSet($vars->tax_id), array('id' => "tax_id", 'class' => "form-control", 'placeholder' => $this->_("Signup.index.field_taxid", true)));
?>
Daniel B's post in Need To Remove Tax Id Number From Registration was marked as the answer