Can I refuse to carry out a variation?

Rarely: most contracts oblige you to comply with a valid instruction and then argue about the money afterwards. A valid instruction is one that changes the work in the way the contract allows, so refusing to carry it out is usually itself a breach, however strongly you disagree about what it is worth, and a breach can expose you to the other side's costs of getting the work done elsewhere. The real leverage sits elsewhere. First, check the instruction is valid at all: was it given by the person the contract names, and under a clause that permits it? Second, check whether your contract lets you object on specific grounds, for example where a change would affect the conditions under which you carry out other, unrelated work. Third, put your position on record in writing before you comply, so your disagreement about value is preserved. The safe sequence is to comply under protest, notify your objection, keep full records of the cost and effect, and fight about the valuation with the work already behind you rather than risk a breach by downing tools.

Say a joinery subcontractor on a hotel refurbishment is instructed by the contract administrator to rip out fitted wardrobes just installed and rebuild them 200mm deeper across forty rooms. The subcontractor thinks the pricing on offer is far too low and is tempted to refuse until a fair rate is agreed. But the instruction came from the named contract administrator under the variations clause, so it is valid, and downing tools would be a breach that could leave the subcontractor liable for the cost of bringing another firm in. The safer route is to comply under protest: a letter goes off first, stating that the work will proceed but the offered rate is disputed because the rework involves stripping out completed joinery, that a full valuation will follow, and that all labour and materials will be recorded room by room. The wardrobes get rebuilt, the records are kept as the work runs, and the argument about value is fought later from a strong evidenced position, with no breach hanging over the subcontractor's head.